The questions I have are, if I want to rewind that section of the coils...
A: Where do I find that ultra fine insulated "magnet" wire?
B: How do I go about figuring out how many turns are required?
A: how fine is it? Magnet wire is available in lots of sizes, from #8 or bigger to #40 or
smaller. In most cases for RF coils the exact diameter isn't important, though it makes
a slight variation in the inductance and self-resonant frequency. You do need to make
sure that the wire can carry the required current, but that is usually only an issue in
transmitters. You can salvage magnet wire from old electromagnets, chokes, speakers,
TV yoke coils, motors, etc. Or you can tell us what size you need and someone can send
you a few feet from their junkbox, since it doesn't take a lot.
See if you have a local shop that rewinds motors and ask to buy a few feet of the size
B: the number of turns depends on the coil. The first step is to count the turns on the
existing coil, which will give you a start, though your coil may have a tuning slug with
different permeability. (You used to be able to order formers and slugs of specific types
If you know that the circuit is resonant on 40m and it has 100 pF across it that gives
you the required inductance of the coil. Then you may need to experiment to find
the number of turns that gives you that inductance on your specific core. You can get
a start using this inductance calculator for air-core coils:http://hamwaves.com/antennas/inductance.html
In the end, just wire up the coil with 100pF across it and use a dip meter to get it
to tune the right range (or a little high, since there will also be a few pF of stray
capacitance in the circuit.)
If there are multiple windings that aren't tuned to resonance, that makes it more
difficult, and you'll have to make an educated guess at the relative impedances
of the different parts of the circuit.