Also in this series: Tiling rectangles with F pentomino plus rectangles
The goal is to tile rectangles as small as possible with the T pentomino. Of course this is impossible, so we allow the addition of copies of a rectangle. For each rectangle $a\times b$, find the smallest area larger rectangle that copies of $a\times b$ plus at least one T-pentomino will tile. Examples shown, with the $1\times 1$ or the $1\times 2$, you can tile a $3\times 3$ as follows:
Now we don't need to consider $1\times 1$ or $1\times 2$ any longer as we have found the smallest rectangle tilable with copies of T plus copies of $1\times 1$ and $1\times 2$.
There are at least 10 more solutions. I tagged it 'computer-puzzle' but you can certainly work some of these out by hand. The larger ones might be a bit challenging.