An unstressed spring has a force constant k. It is stretched by a weight hung from it to an equilibrium length well within the elastic limit. Does the spring have the same force constant k for displacements from this new equilibrium position?

Yes, the spring has the same force constant k for displacements from this new equilibrium position.
In accordance to Hooke’s law, with in elastic limits, tension is proportional to extension from the equilibrium position and always points toward the equilibrium position. A simple harmonic motion is the motion in which the restoring force (F_x(x)) is proportional to displacement (x) from the mean position and opposes its increase.
So,
F_x(x) = -kx
Here, k is the force constant and x is the displacement of the particle from its equilibrium position.
An unstressed spring has a force constant k. It is stretched by a weight hung from it to an equilibrium length well within the elastic limit.
Yes, the spring has the same force constant k for displacements from this new equilibrium position, because for a given spring the spring constant is always constant.