I have run into this underside groove many times. My approach is, I use long lengths of cane for each step, minimizes the tails. Instead of leaving the tails on the undreside, I bring them up the adjacent hole. The next step will hold it in place and I snip the tails off. The binding will cover the cut ends. Much tidier look. See photo. Hope this helps

When you redo the chair with new material if you find the spline is too small for the groove, add piece of cane into the inside of the groove to hold the spline tight.

You may be wise to do a dry fit with the spline in before cutting the excess, adding extra lengths of cane on the inside of the spline where needed to make sure it’s tight. Mark these areas, the. Remove the spline, cut excess, add glue and set spline with extra cane where necessary. This should do the trick.

Hmm..not sure you would be able to stretch it back into the groove and there would be the old glue to contend with. You have nothing to lose. When the groove is bigger than the spline I add strips of cane on the inside of the groove the groove to fill up the space so it sits tight. Clamping can be a pain.

Did you apply a bead of glue before inserting the spline? You should wait 24 hours before sitting on the chair the allow the glue to set, holding the cane and spline in place. I don’t think you can stretch the cane back into the groove now.