The "annual cold treatment" only means it can't be grown as a houseplant forever - it needs to be planted outdoors.
Bulbs produce a single bloom period per season. Once it's done blooming now, it won't bloom again until next year at the appropriate time, generally around mid to late July.

Keep it in a cool area of your home - too much heat will rush the flowers - and enjoy it. Once it has finished blooming, you can cut off the old flowers but leave the stalk. In our zone, you can plant outdoors whenever you like as long as the soil is not too wet.

I've planted many a left over potted lily that didn't sell out in my garden here in Oregon. (I was a florist) I also buy them at reduced prices from local garden centers when the blooms are spent. It is a great way to get them for lower prices. They do really well here. I scored huge pots of Orienpets at Al's last fall for only $6 and they have four bulbs in them. Couldn't buy the bulbs that cheap!

As already advised by Gardengal48, they don't need to be put in the fridge. Just cut the blooms off, leave the stalks and plant.