I'm writing up something I haven't used lately, so
I'll probably leave something out.
[Such as pressing the seams, but y'all know when
and how to do that.]
Leave both shoulder seams unsewn. Sew side seams of garment and lining. Sew the neck and armholes of the lining to the neck and armholes of the garment. Turn right side out, then tuck the front shoulder into the back shoulder (or visa versa, as convenient), pull both out through the still-unsewn hem with the right-side out shoulder inside the inside-out shoulder, sew around the ring to sew up the shoulder seams in both garment and lining.
The above assumes that you want to hem the garment and the lining separately.
It's quite possible to turn a torus inside out through a hole punched in it — when doing my math homework I sewed the cuff of a worn-out sock into a torus and turned it through a gap left in the seam — but when you do that, you swap holes: the doughnut hole becomes the air in the inner tube, and the air in the inner tube becomes the doughnut hole. If, before the operation, you draw a red line around the outside of the torus and a blue line that goes around the torus through the hole, after turning it you'll find the red line going through the hole and the blue line going around the outside.
Or, to put it less verbosely, sewing the lining to the shell at the hem and then turning it right-side out through a gap left in the hem just purely won't work.
But I have a jacket with a clumsily-closed gap in the seam of the lining of one sleeve — there must be a sequence that works for all edges. But it may well involve the zipper down the front.
[Upon reflection, if you cut a torus across, it's just a tube and may be turned ad libitum. But I have no intention of learning how; I never line my bespoke jackets.]