A search for 'Forcing Hyacinths' turns up as many opinions are there are sites.
The best that I saw was the advice on the oldhousegardens.com site.

Most agree that the bulbs should not touch the water; the roots will grow down to the water.
It's also agreed that using a frost-free refrigerator will require some way to protect the bulbs from drying.

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2/16/04 Hyacinthus 'City of Harlem'

My way is to prechill the bulbs in a paper bag in a refrigerator drawer (fruit-free) and
then put them in a cool utility room in vases of some kind that elevate them above the water.

Once they're prechilled, you might be able to keep yours in the garage
in vases in your igloo cooler until you see good sprouts and then bring them inside.
A little blue mold won't hurt;
just wipe it off with a damp cloth when you notice it.

Good luck.
Nell

I agree with the prior response. Also, you can pick up some charcoal pieces (or smash some plain charcoal you already have on hand -- not the kind with lighter fluid!) in gravel-size pieces in most garden centers. A few pieces will help keep the water clean. Don't let the water touch the bulb, and change the water every once in a while.