It isn't universal, perhaps due to the type of silk. I always see abandoned sheet webs for example. From Biology Of Spiders, I read that the webs are pretty high energy internally to make, and so it is efficient to recycle them. As far as the jumper example goes, that is total conjecture based off of 1 observation, so anyways, I'm just an enthusiast, so feel free to check my work, haha. Kinda hoping the main question gets answered more by some pros!

Argyrodes-Dewdrop Spider

They can spin their own webs, but tend to invade and reside in their hosts' webs.[2]

There are some other sorta-examples, but I'm a bit fuzzy minded atm. There is one that tends to build a web attached to smaller orb webs, can't remember the name, but there was little/conflicting information written about them when i looked, however I have observed myself that this tends to be their default behavior, although haven't been able to determine why.

To push a bit further into the lack of source realm, I observed a jumping spider in VA come across an abandoned web, (3", orb-type, close to a wooden rail) ball it up and appear to digest it. I do not know if this was to restock web chemicals or perhaps there was small food I could not see stuck there. I know spiders redigest webs, but after that wondered if at all or how common eating other species abandoned webs was. Different species have different combinations of web material, so surely there would be some incompatible combinations?