Evan Romer's Muffin Puzzle Solution (last updated July 13, 2019)

I can do it with 8/25 as the smallest piece, and I think I can prove that’s the best you can do.
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Divide the first four muffins as follows:
12/25 + 13/25
11/25 + 14/25
10/25 + 15/25
9/25 + 8/25 + 8/25

Reassemble these pieces into:
13/25 + 11/25
14/25 + 10/25
15/25 + 9/25
leaving 12/25, 8/25, 8/25 left over
Do this again, five more times.

So we’ve used all 24 muffins, and have made 18 reassembled muffins, 24/25 each.
And we have six 12/25 pieces left over, which make 3 reassembled muffins.
And we have twelve 8/25 pieces left over, which make 4 reassembled muffins.
Total of 25 reassembled muffins, 24/25 each.
Smallest piece was 8/25.

Proof that 8/25 is the best you can do:

Assume that the smallest piece is (8+a)/25, for some a>0.

Note 0: Every reassembled muffin will be 24/25 of a muffin. And of course every original muffin is 25/25 of a muffin.

Note 1: The assumption implies that every reassembled muffin consists of two pieces only: if a reassembled muffin had three pieces, it would be at least 3*(8+a)/25, which is more than 24/25.

Note 2: If an original muffin has a piece taken out of it that is greater than (9-2a)/25, then the remainder must be a single piece: for if the remainder were in two or more pieces, the pieces would add up to more than (9-2a)/25 + 2*(8+a)/25 = 25/25.

Note 3: a cannot be greater than 4: for if we had a>4, then the smallest piece would be > 12/25, so every reassembled muffin would be > 2*12/25.

Some reassembled muffin has a piece that's (8+a)/25, so its other piece must be (16-a)/25.
But if an original muffin had (16-a)/25 cut from it, this satisfies the hypothesis of Note 2 because (16-a)/25 > (9-2a)/25, so the remainder must be a single piece, so its other piece would be (9+a)/25.
So some reassembled muffin is (9+a)/25 plus (15-a)/25.
But if an original muffin had (15-a)/25 cut from it, this satisfies the hypothesis of Note 2 because (15-a)/25 > (9-2a)/25. So some original muffin is (15-a)/25 plus (10+a)/25.
... and so on ...
So some reassembled muffin is (15+a)/25 plus (9-a)/25.
But if an original muffin had (9-a)/25 cut from it, this satisfies the hypothesis of Note 2 because (9-a)/25 > (9-2a)/25.
So some original muffin is (9-a)/25 plus (16+a)/25.
So some reassembled muffin is (16+a)/25 and (8-a)/25.

But now we have a piece smaller than (8+a)/25, contradicting our assumption.