When I upgraded from mini to midi, I faced that challenge. Took several steps:
1. Jacob's (drill) chucks typically have a mandrel that is MT on one end, and JT-33 on the other (the end that goes into the chuck body). I found that I could purchase a replacement mandrel that would convert my Jacobs chuck from MT1 to MT2 for a lot less than the price of a new chuck. I think I got it from Little Machine Shop.
2. Wine stopper mandrel - if the chuck is MT1, you can purchase a sleeve that has a female MT1 surrounded by a male MT2 taper on the outside. Unfortunately, mine was 3/4-16 female thread that screwed onto the spindle. I opted to purchase 1x8tpi female to 3/4x16 tpi male coupler and it works. That said, I only use that coupler on the stopper mandrel, and buying a new stopper mandrel would not have been much more expensive, and the coupler is almost as long as the mandrel, so it effectively doubles the length of fixing. Wo while financially, this was pretty much a tossup as far as the stopper mandrel is concerned, but it also works on a female-threaded screw chuck that I don't use very much.
3. Pen mandrel - here's where I use the MT1-MT2 sleeve. It works fine.
4. Scroll and collet chucks- the native threading on my chucks is 1x8, and I was using an adapter to fit it to the smaller lathe. So upgrading meant retiring the couplers.
5. Miscellaneous - I'm a big fan of threaded glue blocks and had purchased a 3/4x16 tap to make them for the smaller lathe. I found that I could get a 1x8 tap at Ace Hardware to make the transition to the midi-lathe. They didn't have it in stock, but got it from their warehouse in a couple of days.
There are two concerns that you have to keep in mind. First adding an adapter of any sort can increase runout. I didn't find that this was a problem, but maybe I was just lucky. The other is that no matter which lathe you have, the bed length is what it is, and eventually you will take on a project that calls for a long spindle. Because adapters always move the fixing away from the headstock by some amount, they always reduce the effective bed length of the lathe.