How do you unjam a finger?

Pull it until it "Pops!" For my jammed thumb, I found that grabbing it as deeply as possible with the other hand and pulling out and rotating towards the palm, gave immediate pain reduction & improved mobility. However, one still needs to protect it and reduce inflammation with ibuprofen and ice.

For fingers with swelling, judicious use of an elastic/constrictive bandage will help keep it to a minimum. If there is residual swelling after 48 hours, thumb (or finger) should be regularly & strongly massaged until it dissipates.