Do not——repeat: do not——disturb a blooming bulb. The plant is putting on its show; interrupting it will end that performance.
After the flower has passed and the foliage has wilted (do not remove foliage until then), you can plant the bulb in a larger pot or in the ground. Will it come back (flower again) is another question. 1) Bulbs need a cold spell. 2) While hyacinths are pretty reliable, the "forced" (made to bloom at a particular time) stock of some retail vendors is not as reliable as others.