When working with 3 layers you cant make it inside out and turn it after, due to the sheer weight and bulk.
Lay the curtain fabric face down. Place the interlining only on top turn the sides in and 'slip stitch' them to the back of the interlining only (dont put the needle thru to the curtain fabric). Nice big, non-tight loops about 6" apart. I do a kind of blanket-type stitch for this iykwim?
Work from the bottom up on both sides to avoid any shift in cloth.
Next place the lining on top of this and again, turn under, pin then stitch all the way up to the top. I use a 'long darner' needle for this as its easier on the wrist than a short needle.
All 3 hems are left to hang loose. They are usually made of different materials and will react differently to climate, weight of itself, etc. This usually ensures that there will be no puckering.
You can do a running stitch at the tops to hold all the layers in place for putting on the heading tape. Buy the best you can find.
Very important! Unroll the interlining a few hrs before you cut to allow it to spring back/shrink a bit as it has been stretched on the roll. (Think walls Viennetta).
Tip: I machine a zig zag stitch for the hem of interlining,longest setting. Straight stitch seems to stretch it out of shape and makes it pucker. I zig zag any joined widths too on interlining, right on the edge to cut down on the bulk.
It was a lot more involved in the early days but this method works well. I'll be thinking of you.