Many laptops have room for SSDs in one way or another while still having room for an HDD.
This is the first that I've ever heard of a computer, OEM or otherwise, that you couldn't enter the BIOS of.
OP, if you have a hard drive, then things that you can do to speed it up a little include permanently deleting stuff that you don't need/want anymore, uninstalling unnecessary programs that aren't used anymore, defragmenting your hard drive, and making sure that you don't have malware issues and/or ridiculous amounts of stuff that runs at start up.