Still learning how things work. I appreciate all the help that's been given on this thread. I was wondering, I finished renting some hashing power on the pool a couple days ago. Obviously I solved nothing, but does the power I rented increase the odds of my S3+ that was running on the same Bitcoin address in possibly finding a block sooner while it continues to run on its own?
No. There is no such thing as "progress" in mining, in the same way that 1000 rolls of a die doesn't make you any more likely to roll a 6 on the next roll.