I don't see a temporal progression in the all parables of Mt 13, as if the eight parables, one after the other, told some further progression about the kingdom that the previous one didn't.
One thing that I can see, going by the verb tenses which are used to compare the kingdom with something else, is that, from Jesus' 1st century perspective, the kingdom is already an existing viable entity, not just some distant future hoped for thing. (See also here for a discussion of the phrase, "the kingdom of God/heaven has drawn near.") It is progressively influencing people, until at the conclusion of the system it breaks out forcibly, as it were, into everyone's life. This is the now and yet future aspect of the kingdom. 'Now' starting very small in 29 CE, and 'future' starting at the beginning of the coming great tribulation. (For example, compare Rev 11:15 where God and Christ take over the rulership of the world, leading to the rewarding or punishing of all in Rev 11:18-19.)
What follows is a charting of the verb tenses (in Greek) that describe the introduction of each of the parables in Mt 13. Each parable describes something then already in progress, not just something exclusively in the distant future:
┌ B. Sower (Mt 13:3-9) ("A sower went (Aorist/Ind/Active) out to sow ..."; Mt 13:18 "When anyone hears the word"; Timeless, whenever)
│ ┌ D1. Wheat and Weeds (Mt 13:24-30) (Kingdom ... may be likened to ..."; Aorist indicative passive)
│ │ ┌ D2. Mustard Seed (Mt 13:31-32) ("Kingdom ... is like ..."; Present indicative active)
│ │ │
│ │ │ ┌ D3. Leaven 'hid in dough' (Mt 13:33) ("Kingdom ... is like ..."; Present indicative active)
│ │ │ │
│ │ │ └ D3^ Hidden Treasure (Mt 13:44) ("Kingdom ... is like ..."; Present indicative active)
│ │ │
│ │ └ D2^ Valuable Pearl (Mt 13:45-46) ("Kingdom ... is like ..."; Present indicative active)
│ └ D1^ Dragnet (Mt 13:47-48) ("Kingdom ... is like ..."; Present indicative active)
└ B^ Home Owner (Mt 13:52) ("Every ... instructor who is taught about the kingdom is like ..."; Present indicative active;
. . . . . Compare "every" with "whenever" in the first parable)
On the, "may be likened to" that opens the rNWT's rendering of the aorist indicative passive verb that begins the Wheat & Weeds parable, the older 1984 Reference NWT reads, "has become like"; the KIT interlinear rendering is, "Was likened ... to"; Byington = "resembles"; And here is the BibleHub listing of renderings.
Out of curiosity I also looked up these from my personal library: "Is like" (CBW, Goodspeed, Stern, Lamsa, Barclay, Montgomery); "Here is an image ... of" (Knox); "May be represented as" (Schonfield); "Is likened to" (Wuest); "May be compared to" (Estes, Weymouth); "Has become like" (Darby).
It was the change from aorist to present in all the following that sparked my curiosity. The listing of the eight parables starts with the sower: "A sower went out ..." (Aorist, indicative, active) "and as he sowed (Present, Infinitive, Active) some seeds ..." (Mt 13:3, 4), probably because Jesus had already been preaching for some time, and was continuing to.
The next illustration (Wheat & Weeds) uses the historical aorist indicative passive, maybe because the kingdom is already an established and ongoing 'thing.' It was already building up momentum of its own, thus, the passive. The rest of the illustrations use the present, active verb form, perhaps because they describe the ongoing nature and development of the kingdom from several different aspects.
Anyways, I'm just postulating or thinking out loud on that. Perhaps the verb inflections themselves give a precise picture of Jesus' thinking and viewpoint at that time. Any further comments on that are welcome.
Below are temporal markers within the Wheat/Weeds & Dragnet parables:
Let both grow together until the harvest
In the harvest season . . . First collect the weeds and bind them to burn them; then gather the wheat
The harvest is a conclusion of a system of things
Just as the weeds are collected and burned with fire, so it will be in the conclusion of the system
Son of man will send out his angels ... and collect out ... and pitch them into the fiery furnace
At that time the righteous ones will shine . . .
The sowing, oversowing and growing together occur before the harvest
The harvest is the conclusion of the system
In the conclusion the weeds are bundled and burned first
Then or At that time the wheat is collected into the storehouse
Kingdom is like dragnet let down into sea
When full hauled up onto beach
They collected fine ones but unsuitable threw away
That is how it will be in the conclusion of system; the angels go out, separate wicked from righteous
Wheat/Weeds & Dragnet Combined:
Sowing, oversowing, growing together = Dragnet in sea collecting all kinds of fish
At Conclusion: Angels bind, burn weeds first; wheat put in storehouse = Angels separate fish, throw out bad, put good in containers
It can also be seen that there is a definite symmetry of ideas presented in the parables that are in parallel positions within the chiasm:
┌ B. Sower (Mt 13:3-9) (Sower sows word of kingdom; some respond and bear fruit)
│ ┌ D1. Wheat and Weeds (Mt 13:24-30) (What happens before conclusion; what happens at conclusion)
│ │ ┌ D2. Mustard Seed (Mt 13:31-32) (Kingdom starts small, grows to great size) (Overall view)
│ │ │
│ │ │ ┌ D3. Leaven (Mt 13:33) (Leaven 'hid' in very large mass of dough until whole is fermented) (Overall view)
│ │ │ │
│ │ │ └ D3^ Hidden Treasure (Mt 13:44) (Man finds 'hidden' treasure and secures for himself) (Individual view)
│ │ │
│ │ └ D2^ Valuable Pearl (Mt 13:45-46) (Merchant sells less valuable pearls to obtain one of great value) (Individual view)
│ └ D1^ Dragnet (Mt 13:47-48) (What happens before conclusion; what happens at conclusion)
└ B^ Home Owner (Mt 13:52) (Everyone instructed about kingdom brings out of his storehouse ... [i.e. bears fruit])
The rNWT loses some of the comparative sense between the parable of the leaven and the parable of the hidden treasure by rendering "hid" in Mt 13:33 as "mixed." (Numerous other translation render it like this also. See alternate renderings here.)
Constable's Notes makes some interesting comments on the parable of the leaven here:
THE PARABLE OF THE YEAST HIDDEN IN MEAL 13:33 (CF. LUKE 13:20-21)
This parable stresses the extensive ultimate condition and consequences of the kingdom that would be out of all proportion to its insignificant beginnings.
Some interpreters have understood yeast as a metaphorical reference to evil. However not all uses of yeast in the Old Testament carry this symbolic meaning (e.g., Lev. 7:13; 23:15-18).“Whereas the parable of the mustard seed answers the question of whether the phase of the kingdom planted by Jesus would survive, the parable of the leavening process answers how.”
This parable stresses the hidden internal change taking place in the kingdom between its inception in Jesus’ ministry and its final form when the kingdom will cover the earth in the Millennium (cf. 5:13).
“The kingdom of heaven may be initially insignificant, but it is pervasive.”This fact led J. Dwight Pentecost to call the inter-advent age the mystery form of the kingdom.“The manifestation of the presence of the kingdom in some form in the Church age is clearly taught in the parables of the mustard seed and the leaven . . .”
The fact that a woman put the leaven in the meal is probably an insignificant detail of the parable as is the amount of flour. Three satas of flour (about three-fifths of a bushel) is the amount of flour that a housewife baked into bread for an average family. (But see further on this below. - Bobcat) 
“Practical applications of this parable to present readers can include the following. First, believers should depend on what God is doing through His Spirit in the present age. Second, Christians should be suspicious of any man-made, externally influenced institutional structures that say they are the manifestation of God’s kingdom. Third, believers must be cautious about setting dates and presuming the arrival of the kingdom since the parable gives no hint as to when the permeation ends [other than to say that the mass does get fully fermented; for comparison, see my comments about the 6th trumpet & 6th bowl below - Bobcat]. Fourth, Jesus’ followers can be confident that regardless of any current perspectives, the kingdom of God has a glorious future.”
That last paragraph just above was especially interesting.
On a side note in connection with the leaven fermenting the whole loaf: According to R.T. France in his NICNT-Matthew commentary, the 3 measures of flour is a huge amount of dough. France describes it as enough dough to feed a small town. In the 6th trumpet in Revelation 9, mankind is plagued with the hopes of causing a repenting. But the description of that trumpet ends with mankind, as a whole, refusing to repent. (Rev 9:20-21) In the parallel 6th bowl, the action ends with 'the kings of the earth and their armies gathered at Armageddon (i.e. the mountain of Megiddo), which places them in a threatening position in relation to Jerusalem and the land of Israel (Rev 16:16; I am understanding symbolically). The whole of mankind, outside of God's servants, has taken an oppositional stance to God. This compares well with Jesus' description of how the leaven eventually ferments the whole mass of dough. Either for good or for bad. At that point it becomes the appropriate time for the 7th Trumpet to be blown (Rev 11:15-18) and 7th Bowl to be poured out. (Rev 16:17-21)
(See here on the parallel nature of the trumpets and bowls. And here for the temporal indicators that determine the order of the trumpets. WT has trumpet 7 occurring before trumpets 5 & 6. Revelation indicates otherwise.)
Index to Parables