Another approach, if you don't need it to be truly random, is to take the same random output from the object info node, pass it through math.multiply and multiply it by some quite large value (100?), then through a math.modulo with modulo 1.
I think for almost all purposes this is going to seem like a brand new random number. At least, when you wouldn't notice a minute difference of the main random number. We're essentially chopping up the digits of the random number we were given into chunks, and using each chunk as a different random number.
So say we got 0.9172352114 as our random number, we can use that is is for one purpose, then multiply and modulo it to get 0.72352114 as our second. We could do it again and get 0.352114 is our third one.