Is it possible to postponed the Turn 7 reshuffle to Turn 8?
I've had many games without a turn 7 reshuffle. It's totally possible if playing with the 7 optional cards. This depends on the following:


How many cards are removed from the game in Turns 1-2.
How many cards are held in Turns 3-6: players may hold:


Multiple cards by playing The China Card, Ussuri River Skirmish/Cultural Revolution (multiple China card plays), SALT Negotiations;
No cards through the plays of Blockade, Five Year Plan, Grain Sales to Soviets, UN Intervention.
The space track may also throw a wrench into that, but we won't investigate it as impractical.

Whether Ask not what your country can do for you... or Our man in Tehran have been played or not. This technically falls into #2, but I consider them different enough to deserve a mention on their own.


Here is a theoretical example, assuming: the 3 aforementioned factors work in the way of delaying the reshuffle:


No events are removed in Turns 1-2.
No China Card play in Turns 3-6.


Turns 1-2

There are 38 early war cards: 3 scoring, 6 neutral, 14 US, 15 USSR. For the sake of simplicity, let's also assume no China Card play or hand-size reductions in Turns 1-2, albeit you shall see that it does not affect our analysis. Thus, we assume that players hold a single card in both turns.


16 cards will be drawn on Turn 1, leaving the deck with 22 cards.
14 cards will be drawn on Turn 2, leaving the deck with 8 cards.
At the end of Turn 2, there will thus be at most 14+14=28 cards in the discard pile, assuming that no starred events were removed.


There is no way to trigger the first reshuffle earlier or later than turn 3. On the one hand, there are at most 3 hand-size reductions (Blockade, Five Year Plan, UN Intervention), which would leave the draw pile at the beginning of Turn 2 with 5 cards, if all were played on Turn 1. On the other hand, only 2 extra cards may be held, during Turns 1-2, still not enough to delay the Turn 3 reshuffle.

Turn 3

When dealing cards on Turn 3, there are:


8 cards in the draw pile
14 cards that have to be dealt
28 cards (at most) in the discard pile


Therefore:


The first 8 cards are dealt to both players.
The (max) 28 discarded cards are reshuffled to form a new draw pile.
Finally, 6 more cards are dealt to each player, leaving the deck with 22 cards.


Note that prior hand-size reductions or plays of the China Card in Turns 1-2 do not affect the aforementioned numbers. If players hold more or less cards, this will of course affect which cards are held through the reshuffle, but not the total number of them. For instance, if the US cannot hold a card on Turn 2 due to Blockade, then at the beginning of Turn 3 there are 15 cards to be dealt, but also 29 cards in the discard pile, thus not affecting our numbers at all.

Let's assume now that both players will now on hold exactly one card, thus holding 1 card at the end of Turn 3.

Turns 4-7

There are 48 mid war cards: 4 scoring, 11 neutral, 18 US, 15 USSR. On turn 4:


There are 22 cards remaining from the early war.
The 48 mid war cards are shuffled in the draw pile.
A new pile is formed of 70 cards in total.
We assume that players hold a single card and thus 16 cards are dealt each turn.


This would mean that at the beginning of Turn 7 there will be 70 - 16×4 = 6 cards in the draw pile, thus not triggering a reshuffle! This would lead to the 23 late war cards being added in Turn 8, leading to a delayed Turn 9 reshuffle. This would mean that all late war cards would be guaranteed to be drawn by the beginning of Turn 9.

Hastening the reshuffle

The aforementioned analysis suggests that 6 cards need to be "eliminated" to trigger a Turn 7 reshuffle. These could be:


Early war starred events eliminated in Turns 1-2, thus not entering the reshuffle.
Hand-size reductions during Turns 3-6, increasing the number of cards that need to be drawn.
Playing Ask not what your country can do for you... or Our man in Tehran, which either thin the draw pile or cycle faster through it, respectively. These may also be played on Turn 7, not just during Turns 4-6.


On the other hand, not removing many starred events in Turns 1-2 and successive plays of the China card in Turns 3-4 would result in not triggering the Turn 7 reshuffle. Playing SALT negotiations lets you hold multiple cards, thus delaying the reshuffle as well.

Fine notes: if the reshuffle has not been triggered by Turn 7 and the US player holds Our man in Tehran, then they have a huge control on what will enter the reshuffle or not, if there are at most 5 cards remaining in the draw pile. In this case, he should:


Play as many US cards or scoring cards he would like to enter the reshuffle again.
Play Our man in Tehran to trigger the reshuffle.
Play Ask not what your country can do for you... to discard the USSR or scoring cards they would like to miss the reshuffle (or just space the former).