Struggling to find a quiet spot for my noisy spinning wheel in a small apartment

I recently picked up a vintage spinning wheel to try my hand at making yarn, but my apartment is pretty small - just a studio with thin walls. The wheel makes a constant rhythmic clicking that echoes way too loudly, especially when I spin in the evening. I've tried placing it on a thick rug and even draping blankets nearby to muffle the sound, but it still carries through the room and sometimes bothers my neighbor. I want to keep practicing without annoying anyone or feeling self-conscious, but space is so limited that moving it to a separate room isn’t an option. Has anyone dealt with noisy craft tools in tight living spaces? What are some creative ways to deaden or redirect sound without building a full enclosure or investing in expensive soundproofing? Would something like a portable folding screen covered in soft fabric help, or am I better off timing my spinning sessions differently?