Just my opinion, but if I was buying a new chuck, I'd get one I could mount on the spindle without an adapter. The reason is that the more "stuff" you use with the chuck, the more chance for runout (not spinning true). I'd use the adapter for the chuck you have, but I wouldn't buy another one to use with the adapter. At least not before I tried working with the adapter with the chuck I had to see if there were any issues.
Go into the world and do well. But more importantly, go into the world and do good.