|Ahhh... the swimming pool is my poison ivy rash treatment :-). The chlorine will help dry up any rash and the sheer amount of water in a pool would likely dilute any urushiol oil that may be in the water.
The question in my mind is were you cleaning up after a contact with poison ivy plants or was this after and you just had a rash? There's a big difference here as if you've already cleaned up and now are dealing with a rash, you really can't reinfect yourself unless you weren't diligent about clean up. The oozing from a rash is not urushiol oil and certainly won't cause any spreading.
Going into the hot tub could be a problem as a rash is open sores and hot tubs are notorious for being the perfect growing medium for pathogens.
So - the simplistic answer here is not likely.