Old SAT and New SAT Superscore?

If I took the "old" SAT and did well on one part, will that part "carry over" and combine with parts of the "new" SAT for a higher score?

Probably not. The changes to the content of the SAT mean that the different sections are no longer measuring the same things - the Math section tests different skills, the Critical Reading and Writing sections have been combined into one score out of 800 (instead of two scores out of 800), and the essay is now optional, is a completely new task, and is scored on a different scale. As Allen points out in is 2/26/2015 comment to this article, schools might superscore within each version of the test, but not between them.