Can I use the leftover pickling brine?

Not sure any of the above have read your question completely. You are asking; can you store extra brine that you made but was not used in your pickling process. I do this systematically when my Green Beans are going crazy. I mix up extra brine and store it in the refrigerator for up to 2 weeks. The brine is simple pickling salt, water and white vinegar. Then I can use it incrementally as my produce comes in. Saves me a few steps by simply reheating premixed brine instead of making it each time. I like this approach because it allows me to put up a few pints at a time while the beans are fresh. Have never had a problem with my Dilly Beans.

Yes, if I comprehend correctly, it was part of the brine solution you heated, but it was not actually used in pickling. There is (arguably) some concern with reuse pickled liquids, on the basis of potential levels of bacteria.

Personally, I have reused pickling solutions - though these have mainly been vinegar solutions based and not brine solutions. Have yet to delve into natural fermentation 'pickling'.