The usual period solution to a demand to increase production seems to have been to throw more workers at it, though. Easy access to lots of slaves (or just cheap labour) tends to do that.

Plus, there's the question of fuel. Teh energy issue. Steam engine is worth its weight in scrap metal if you don't have something to burn in it, in practice meaning wood and coal. Neither of which the Mediterranean region was all that blessed with, AFAIK...
Have fun making primitive steam power cost-efficient (which is something of an absolute precondition for it becoming a factor in the economy rather than mere curiosity) in that situation. Heck, I doubt if period metallurgy was up to the task either - do recall that by the time Europeans started dabbling in proto-industrial production techniques around the Late Middle Ages or so (using water power), they had some rather massively more advanced metalworking know-how to "build the tools to build the tools" with.