I learned to turn on a metal lathe so i do things different from most wood turners.
In my 4 jaw chuck I don't have individual control of each jaw as I did on the metal lathe so I use shims to center the work piece in the jaws. I set up a one inch travel dial indicator to tell me when the piece is centered best. Since I am using shims I will tighten the chuck and tap it several times as I check to be sure it is tight and centered.
Use a jam chuck so you can mount the bowl between centers to true the tenon. Then you can get a good safe grip on the fresh tenon with the chuck to finish the bowl.
X2 what Ted said. I use a 6" long, 4" in diameter PVC pipe, with some leather glued to one end, put that in my chuck, instant jam chuck. Place the bowl, bring the tail stock up and return the tenon.
I do like the others. Make sure you leave the center point dimple when you turn initially. This is the last thing to come off for me after all turning is complete and the foot is being cleaned up. Depending on your lathe and the depth of the bowl/object you can just put a piece of leather (or other thin tough material) over the chuck jaws and use that as the friction drive. Some people do not even put leather over the jaws since the inside has to be returned also. It doesn’t take a lot of pressure to drive it if you are only reforming the tenon/recess and maybe an inch or so out from it. I’ve never had one come loose.
If the inside is still rough, you can just jam it against your open chuck jaws. It typically stays still long enough for you to re-true the tenon. At least that works for me...