Alum lasts a long time unless...

Bon Jovi "Slippery when Wet" springs to mind!

Anyhoo......with a waste not want not mindset don't be throwing those bad boys out. Get yourself a plastic pump action spray bottle. Break the bits into sizes small enough to go in the bottle. Half fill the bottle with water. Replace top and shake like a goodun!
This will last for.......well possibly forever with each time you run out of water you just top it up again.
Also a handy item for the travel bag.

Thanks for the suggestions everyone! I got some replacement alum blocks for $3. They look a little dirty compared to my pristine Osma...

The chunks of my old alum will still be used. I'll just file down the sharp ends.

Maybe you could grind it to a powder and use it that way. Maybe applied with a dry shaving brush. Just an idea.

I use a roll on alum, and spray bottle alum also. Non breakable.

http://www.naturallyfreshdeodorantcrystal.com/

Alum-On-A-Rope?

Same thing happened here. I just ended up buying a new one as it was too much of a pain in the arse to use the broken bits.