Some alternatives to consider:
As Adam mentioned, you can actually just put the leaf in the cup. This is actually a very easy way to get a great cup. For green teas you might want to use a glass instead of a mug.
You could also use your current 20oz pot and only fill it half way (or however much you want to brew at a time - you don't need to fill it to the top), or get a large infuser basket for the teapot that would still leave lots of room for the leaf to move around and that you could remove when done steeping.
Other than that you might seriously consider getting another pot. You can get an infuser mug for as little as $5 at your local department store, or you can get a nice pot for $10-$20 (some are even less than $10).
I used an infuser mug for a long time as my primary brewing vessel, and still use it for some things (that or a ~8oz pot that I picked up for around $5).