As you describe it, the method would be a narrowing of the existing method but, due to added mechanism, would have broader use than the previous method. It is not really a generalization of the old method.
There are similar cases where patents are awarded for someone finding a sweet spot in an already known method. Maybe at a certain temperature range a process becomes, unexpectedly, much more efficient and therefore has broader applicability. But it is still a narrowing of method.
Separate from the issue of prior art is the issue of freedom to operate. If the prior art is subject to an in-force patent, then you will have a patent that you can't practice without the permission of the prior art patent owner. They have patented doing steps A, B and C, You have patented doing steps A, B, C and D but to practice your patent one is still doing A, B and C.