Hi Amber, as far as I know once the foliage of the forced bulbs have fully withered it can be removed already. And from what I have read most sources recommend just getting a new set of bulbs come Fall season since forced bulbs may lack enough energy to rebloom again. But you can certainly try planting the bulbs outdoors later, Hyacinths need that particular cooling period too.
I have not forced Hyacinths in water nor in gel beads, I have planted some Hyacinths bulbs in containers outdoors and so far they do return every year but their size is not as it was before. So far what I have been able to force are Paperwhites and Amaryllis bulbs. But still a learning curve for me..the Paperwhites I got only one bloomed out of 4
So the one that has bloomed already, I am just letting it sit in the water gel bead set-up, while I have planted one dud outdoors and the 2 other duds I have left in another water gel bead set-up. So far the last 2 are making roots, no blooms..oh well...
There is a forum for Bulbs here in ATP, I am sure if you post your question there as well, they can give you a lot of insight about Hyacinths.