Essentially you are dealing with fairly primitive hot air engines, and the splitting between two cylinders is similar to early Brayton "Ready Engines". Indeed, atmospheric engines were described by Hero of Alexandria as far back as the First century AD, so this type of technology has a long history.
The limiting factor with any sort of atmospheric engine is there is a hard upper limit to how much power you can produce. Even if you could somehow arrange for a perfect vacuum, the maximum force pushing on the cylinder is 15 PSI (101.325 kPa). You would either need an improbable number of cylinders or really huge ones to generate significant force.
Adding extra stages as you suggest would improve efficiency somewhat, but there would be additional parasitic losses from the extra linkages and the airflow through the cylinders and heat exchangers. Without the actual design it is impossible to calculate how much loss you will get, but at some point it is fairly certain that the losses will outweigh the potential gains. With primitive, hand made machinery this will become a very large effect very quickly.
So while it is possible to build atmospheric engines and add additional stages to increase efficiencies, there will come a point that the extra machinery is actually drawing more power from the engine than it is contributing. Unless there is a very compelling need to gain extra power from the engine, economics suggests that the advantage will go to the most basic engine capable of doing the job, since it will be cheaper to build and operate.
