I haven't use HF recently but in the past I learned that HF drill bits and taps weren't good quality. Buy the tap you need. I understand a good tap and die is expensive, and get yourself a set of tap sockets.
https://www.tooldiscounter.com/product/lisle-socket-set-tap-lis70500
You don't need to re-drill the hole if you are just chasing the threads with the same tap as current threads.
You would need to drill for creating new threads in a larger hole or any new.
Mr5 is correct there are different taps for different jobs. starting tap is for creating new threads in a new hole, A bottoming tap will cut threads to the bottom of a blind hole. This tap is also good for chasing threads to clean or correct a burr, but not good for starting new threads.
Most tap and die kits come with starting taps, in most instances you can use to clean threads.
When cutting new threads use the correct size drill, normally stamped on tap. Start the tap straight with hole, be careful not lean it while turning.
Cutting fluid lots of it. After the tap makes it in the hole 2-3 whole turns reverse the tap 1/4 to 1/2 turn, this breaks the curly ques and makes it easier to continue, do this every 1/2 to 3/4 turn as you continue. you may want to remove the tap and clean the flutes several times.
Be very careful if a tap feels like it doesn't want to turn don't force it, stop and investigate. Taps are very hard, so much so they are more brittle and will snap. you can't drill and extract a broken tap.
Also for cleaning threads on stuff you take apart and assemble a thread chaser kit is nice. not really meant to cut threads and not as hard as a tap, so can be removed much easier.
https://www.tooldiscounter.com/prod...MIh_L6nIH46QIVISCtBh0dpQsLEAQYAiABEgLZ5vD_BwE