With the chuck mounted on the spindle, close the jaws to the smallest diameter. Take the untrue bowl and place it over the closed jaws of the chuck. The jaws should rest in the bottom of the untrue bowl. Bring up your tailstock and centre with the previous centre mark. If you do not have, or did not mark the previous centre (where the tailstock penetrated the bottom of the bowl during roughing), you will need to centre the bowl visually. This may take a few mountings to get the bowl near the correct centre.
With the bowl mounted and the tailstock centre brought up, tighten the tailstock. The bowl will be held by the friction (and pressure) between the chuck jaws, the bowl and the tailstock. You can now bring up the toolrest and return the dovetail (or straight boss) with a small skew chisel. Remember to keep your speed low, whilst retruing the tenon boss. I usually keep the revs around 500 or so during this operation.
Once the boss has been retrued, simply remove the bowl and remount it inside your chuck jaws in the usual manner. I typically take a few light passes on the outer bowl wall, to remove some of the warpage on the outer skin before I remove if from the drum chuck mounting. If the untrue bowl is too deep to mount over the closed jaws of your chuck, you can turn a "stubby" tennon that is long enough to clear the bowl rim. Simply take a wide/deep piece of scrap timber and turn a tenon on it. On the end of the tenon, turn a slight concave. Then, mount your untrue bowl in the same manner as above. The extended tenon acts to hold the inner bowl bottom, whilst you retrue the tenon boss.
Remember, when roughing green bowls... Leave an oversize tenon boss on the roughout, so when you retrue the boss after it has dried, it will still fit your chuck. Another way to remount untrue bowls for retruing the tenon boss, is to turn a dovetail boss on the inside of the bowl when you rough it. Although this will also warp a bit, you can usually get a good enough grip on it to quickly retrue the outer (primary) tenon boss.
Remove the bowl from the screw chuck, reverse it and mount on to your chuck jaws. You then remove the bulk of the material from the inside, to the wall thickness you deem appropriate, BUT you leave a pillar of wood in the centre with the screw chuck hole still in it.
Put aside to dry/warp/split/crack, hopefully not all of them, then when you remount just screw your screw chuck back in place and true the base mounting up again for a concentric grip. You can either true it up or make a new one if you've left sufficient thickness. Keep the revs low till you've got everything in balance, the pillar could snap if you have too much vibration going on.
This is less practicable if you are making small diameter bowls as you won't have much room to work between pillar and bowl wall but anything over 8" dia should be okay.
When you resurrect the dried and distorted bowl, remount using the inside spigot so that you can then re-do the outside, and then reverse as you normally would to do the inside.
This works well and the spigot on the inside doesn't seem to afffect the drying.
In fact the additional mass tends to reduce the splitting in the foot area if a particular wood has a tendency to do this.