So there are some really cool plant fiber yarns other than cotton, like bamboo, banana, soy, cellulose, linen… There are also the more common acrylic and cotton yarns.
I haven’t tried all of these on their own but I have read about a lot of them and tried a few.
I tried bamboo, which is especially known to have non- allergenic properties, and it’s really soft and nice! However, it’s not very sturdy on its own, so imo it’s best in a cotton blend. The more bamboo there is in the blend the softer it is, and the more cotton in it the stronger it is.
Banana and soy fibers resemble silk, and can be used as an alternative to it, both are also known to be very soft.
I don’t think I’ve ever tried Linen on it’s own so I can’t say, but it’s said to be very durable…
Most plant fibers in general are softer than animal fibers bc hair/fur/wool all have scales to it. These make it prone to sticking together but it also makes them more rough, itchy and uncomfortable to wear.
I think going through stashes on raverly as was suggested before is a good idea. I usually also just use a bunch of filters on online yarn shops to only get the kinds I want 
Of the brand yarns I’ve used I really like King Cole’s bamboo blends, Kartopu Punto(!!), and Madame Tricote Paris’s cottons and acrylics (I can check the rest of my stash next time I’m home since I don’t really document it anywhere 
I’m also always on the lookout for indie brands so here are a few sellers I like or want to buy from (unfortunately most of these are a bit on the pricey side):
BlurBarnFiber- they make a lot of unusual types of yarn, always in good quality and beautiful dye. The item descriptions in their Etsy are very detailed so you know exactly what you’re getting (I now see that they even made yarn from steel?? That’s probably irrelevant to your project but I just saw that now and I think that’s really cool)
BirdiesKnits - lots of hand dyed 100% organic cotton yarns, absolutely lovely dyes and good customer reviews
The Spinnacle- beautiful hand dyed yarn, a lot of it is wool but I talked to her and she says she also has a good soy silk base, which I can’t wait to check out when her store reopens. You can find her on Instagram by the same name
Www.thespinnacle.etsy.com
I like buying from indie brands bc I want to support independent artists, and because hand dyed yarns can be really unique! Plus, handmade things feel so special. … But they are usually way too expensive to rely on as your primary source
it is nice however to get some special yarn for a special project once in a while 
Hope this was helpful! If I was at home I could probably find you some more specific examples that I liked, but hopefully this will do for now.
Lmk if you find anything special 