You could attempt to cut its roots out of the hydrangea's root ball. It would be easier if it were indeed a weed, but it could be a tree sapling, in which case it might be more difficult. Try using a hand spade, but it might require a small hatchet or some pruning shears. As long as you get the woody crown and/or most of a taproot out of the ground, the nuisance should not return, or at least not for some time.

Try cutting the stem back and immediately painting the cut with a strong weed killer, be sure not to drip any at all on the hydrangea. I wish I knew my weed killers to be more helpful in that area. You may have to repeat if you see new growth, always on a new cut.