If you never saw your primary care doctor
after jamming your finger, you should do so now. The one finger injury
that sometimes requires surgery
is a broken bone. Granted, if there was not major deformity of the finger or pain after you injured it, then it is unlikely that you broke it and, furthermore, it would be too late to fix many minor finger fractures at this point. Nevertheless, your doctor will be able to examine the finger to make sure there is no major problem and may take an x-ray to rule out a poorly healing fracture.
I think more likely, however, is that the sprain you sustained (which is what a 'jammed finger' is) is still just healing. Most people don't entirely appreciate how long it takes for a sprain to heal, and I think it is totally reasonable to still be experiencing some mild discomfort even 1 month out. This should gradually continue to get better with time, as long as you are careful and don't reinjure the finger.
Again, please talk to your primary care doctor about the issue; he or she will be able to answer any additional questions you might have.