Use thin material, like pigskin, or other choices you have available for the lining in the spine (rings) area. Glue it down solidly. It will still pucker a little.
Another option is to groove the spine on the inside where you want it to fold. And again use thin leather for the lining. Make sure you get your glue down into the grooves and use a boning tool or similar to work the lining leather down into the groove. You will have a sharper fold, and less noticeable puckering inside.
While gluing, in both cases, fold the cover about half way closed and let the glue set before opening or closing the cover.
Yet another option is to use lighter weight leather for both the inside and outside, with a stiffener, like cardboard, between the two. Need the stiffener cut in three pieces, two for the front and back (same size) and one piece for the spine. Leave about 1/4" or more space at the folds. Look at the typical vinyl covered loose leaf binder as an example. Pull an old one apart, might even be able to use the board as your stiffener.
Try it out on some small scraps to see what you like the best.
Tom