That’s my method as well and its saves a lot of twisting.
I’ve got 3 pinvises and one of them will not hold the smaller bits. The tape trick is a good idea, I’ll have to try that sometime.
Or, you could use one of the mini chucks from Micro Mart, unpowered like a pin vise or powered with a small drill motor. DJ.
Look in the Micro Mark catalog for a small chuck that will fit into your pin vise. These chucks will close down to zero. I have two Blue Point pin vises and I keep the chuck in one of them and use the other for collets.
I just got a new Starrett pin vise for 0-.055" bits and now I see why they cost so much. 100 times better than the General double end one I have been usiing for the past 30 years. Highly recommended! You get what you pay for!
-Bob