How do I use the smallest amount database file space to rebuild an index on Mircosoft SQL Server Express due to the 10GB limit
The reason is that index rebuild means creating a new index under covers and then dropping the old index. I.e., while the operation is running you need twice the amount of storage available. If that isn't doable for you, you have basically two options:

Do it yourself in the opposite order: Drop first and then create.

Or do REORGANIZE instead of REBUILD.