I don't know if there's an easier way to go about this (other than drag 'n drop, of course), but if you would rather search through your files, you can hit the paper-looking icon in the top left, then hit "import", and select your costume.
It should then get imported as a different sprite, then you can just drag and drop the costume from that new sprite over to the desired one.
Drag 'n drop is much easier though, so I'd recommend you continue using that.
"Import" checks the file type, so you can import sprites, costumes, sounds, libraries, text, ...
(You export a sprite by right-clicking its icon in the sprite corral. The exported sprite includes not only its costumes but its sprite-local blocks, its scripting area, the works.)
PS About the button in the costume tab, yeah, I agree. But it's not urgent so don't hold your breath.
It is not working for my Snap! I am using a tablet. I am trying to upload multiple alonzo costumes but when I press import and choose the file, it is not working.
Is there a problem with my Android Chrome, or is it another issue?
You trust level is a new user, which has restrictions on posting images. After you have been on the forum for a while, you should be able to become a basic user to post images.