Here's my best guess, early model 44's have an adjustment screw for headspace in the breech block that is visible when you open the action. Possibly someone has backed out the screw so it locks up tight, but unfortunately it increases headspace, along with not allowing the breechblock to travel upwards for the firing pin  to align properly. . Here's a picture of an early model 44 breech block that has the adjustment screw.

What does the spacing look like when you have a case in the chamber? What happens when you loosen the screw?

Learn something new today!  Did not know the early guns had this screw.   

So, if you turn it in to raise the breech block…how about a longer screw… or maybe the tip is worn down?
Bob

As with most "as-found" Model 44s, the toggle linkage is loose from wear and firing.  Solution is to ream all the toggle pin holes to a snug fit on 5mm pins.  Also possible but unlikely that the lever pivot is loose.  p/m me and I'll send you the detail setup procedure for tightening up your action. 

In a properly set up 44 the toggle linkage clamps the breechblock hard against the barrel face.  That screw in the breechblock was a coverup for the fact that as the gun ages the loose linkage fails to do so, so the lever "droops".  A later coverup was a springloaded plunger in the lever.  What Stevens should have done but never did was to make the link out of a high carbon steel, through-hardened.  Instead, right to the end, it was a mild steel stamping, lightly case-hardened.

I do have a little kiln that I have used for springs, so I could do links from annealed tool steels and harden them, but I haven't.  Instead I've done two 44 links from 4130 prehard, and a couple for 1915 Favorites from 4140 prehard.  Both will cut pretty easily with carbide.  The milled profiles aren't exact, since I don't have CNC.  I round the exterior contours on a belt sander.   I will never live long enough to determine how much the service life of my .25-20s will be increased, but it just has  to help.  I have some gunsmith's mistake of a .32-40 (rebarrelled a gun without the lugged hammer) to do this summer if my health holds out.   

Something I've found interesting is that the hole spacing of factory links varies. I've made up a set of dummy links with varying spacing to test a rifle before committing to making a final example.  (I read that Ballard rebuilders do the same.)  So it won't likely come to pass that a parts house will offer a generic link.  I'm more than half convinced that Stevens was doing a "selective fit" process when assembling the rifles.

Just case hardening a soft steel link is the problem.  The link takes a load of about 75% of the breech thrust on those little bitty pins, and the soft core is not strong enough to resist the battering.  The holes get pounded oval, and the breechblock is no longer held up as high as it should be anymore.  Reaming will bring it back to battery for a while, but eventually it will happen again.  What it always needed was a steel with a higher yield strength.

In wildthing's OP, he just mentioned that the firing pin hit low, he didn't say by how much or what caliber.  He did not mention if his rifle was a bought as a center fire or rim fire.  The breech block shown by Sure Shot looks to me like it is a 44 rim fire breech block.  Another member here told me that he bought a 44 rifle that had a center fire breech block that someone had screwed a .25 RF barrel into, which worked out great for him when converting Hornet brass to reload his .25 Rf as a CF.  Could it be that wildthing has CF barrel with a RF breech block?