You're spot-on. The free floating symbiote was held prisoner by the Fantatic Four for awhile but escaped and stalked Spidey. For a few months afterward he did have a cloth black costume. When the symbiote found Brock and became Venom Spidey went back to his original threads (thanks to fan outcry at the time actually).
it's been a few years since i read the comics..hell it's been more than a few years since they even came out, so correct me if i'm wrong, but i remember a short time where Venom was 'brand new' and Spidey was still in the (cloth) black suit, and Venom started out doing petty crime type stuff to turn public opinion against Spidey. I also remember Venom kidnapping Mary Jane. From a story-line standpoint, that was the reason Spiderman went back to the old Red & Blue suit