Funnily enough, I have cut up many old tshirts and balled it up into yarn and knitted bath mats with it - they are fantastic and wash well too! Now I’ve cracked crochet I’m thinking of doing a round one with tshirt yarn. You could always dye some of the white tshirts first to add a bit of colour or knit it and then dye it.
As far as picking up more tshirts - ask around friends who may have some getting thrown out. Also it may be worth checking your local charity shops and asking them if they have any old tshirts that aren’t good enough for them to put out in the store - they usually sell these as scrap fabrics that get recycled elsewhere but they may be willing to let you have a bag for a small donation.
I’m terrible for keeping fabrics - they’ll be useful for something one day - right? Cotton fabrics get made into bags here, I keep jeans for patches too and other projects. I keep pieces of old voile/plain net curtain to use in wet felting. I’ve recently seen a demonstration of some fab dyes for fabrics that are mostly synthetic so plan to get some of those to re-use any synthetic fabrics, ribbons, etc and give them a new lease of life as something else or bits for making cards, etc.
Good idea for a thread, look forward to other inspiration x
ETA: if you make tshirt yarn - cut the tshirts on an angle so you can create one continuous piece to ball up - less joins than cutting horizontal strips and it curls up nicer too.