What type of stabilizer are you using?
For some stabilizers, I cut out a window and replace the window of stabilizer. I use a glue stick to hold it in place. Make sure the window is secure as that area can be the cause of issues with either tearing or moving.
You can use 2 pieces of cut-away glued together. Hoop them and then cut the window for the embroidery. Add the actual piece of stabilizer (sticky-back) and proceed.
I have used a glue stick and put 2 pieces of no-show-mesh together if I need to extend the stabilizer to fit the hoop. I use this type for most projects as it can be left in the quilt block or project easily.
Next tip up is to use WSS double-sided tape to put the pieces of stabilizer together. Use enough tape along the seam so it won't pull apart when stitching. My concern is the area where the embroidery foot is going to be moving along and to prevent it catching as the foot moves. This works well to join 2 pieces together.
Next tip is to cut the stabilizer so you can do multiple hoopings. Just leave the block in the hoop and then move the hoop so you can make the next block.
I don't stress over saving stabilizer but try to hoop wisely. The LQS always has a sale come along and I stock up then.
One how-to below:
https://www.youtube.com/watch?v=9-I3DgGFHUs