If it's an inch thick at the handle, I would add another piece of osage about 3/4" thick, and that would be enough to make the kind of handle I prefer. I have done this by perfectly flattening both gluing surfaces, run a toothing plane blade over them to groove their surfaces, glue and clamp them together with Smooth On epoxy, cured with a shop light next to it for warmth, and then treat it as a single solid piece from then on. This is a good, strong joint, but additionally, a long gradually sloping dip/fadeout will help to slow the bend before the handle. Many of those that fail aren't glued this way and the fades ramp up too rapidly.
This is no different, and actually LESS in danger of coming loose than the handle pieces that are glued onto bamboo backed bows, and trilams, and such, which are 1/2" or less before the handle piece is added. I prep them the same way... with no semblance of a 'pedestal' or power lam, and I never have issue with them either. It works... even when it bends some into that glue joint.