" I turn the heat off after the heating season, leaving the furnace
running for
the hot water.......The hot water, currently runs off a small tankless
unit, which does
tend to run out of hot water on them frequently. "

This is confusing. Is the current hot water supplied by the furnace or
a seperate system?

In any case, I would agree with the comments that an old coal furnace
converted ot oil is likely very inefficient compared to today's
systems. I would seriously consider replacing it. Also, I would think
that a seperate gas water heater is going to be more efficient than
using a furnace to heat the hot water. In the winter, when the furnace
is running to generate heat too, it may not make much difference. But
I believe firing a larger furnace, especially an ancient one just to
supply water is going to be a losing proposition.

You could also consider an on demand hot water gas system.