Hillary wouldn't have run against Gillibrand. She would have moved to a state with an open Senate seat (Arizona) and run there, and then, if she won, tried to make Gillibrand's life a living hell every day on the Senate floor.
Or she would've tried to take out a Republican incumbent, helped her party and tried to make Gillibrand's life a living hell every day on the Senate floor (For example, she probably could've beaten Dean Heller in Nevada.)
I don’t think Hillary could have successfully ran as a carpetbagger in a purple state like AZ or NV. You might say “She did it in NY in 2000!” but that was different for a number of reasons: The state was solid D, Bill was the popular incumbent president at the time, and it was as good a home for her as any — let’s be honest, she was never really an Arkansan, so the only state that could have been more appropriate for her to move to and run from is her birthplace and childhood home of Illinois. Now that she’s lived in New York for 20 years, has lost a presidential election (twice if you count 2008), and the Clinton brand doesn’t carry as much weight anymore, her moving somewhere else to run would likely not work out so well.
Also, I don’t think Hillary is quite as sadistic and vindictive as you make her out to be even in this imaginary timeline. If she ran, it wouldn’t just be to get revenge on Gillibrand, let alone to torture her.