I have a clock radio that gets lousy reception in my office at work.
I want to experiment with adding external antennas to it. Maybe a
half-wave looped dipole for FM and a small external loop for AM.

My question is how to add the external connection.

The radio has a loopstick ferrite antenna (I presume for only AM) The
loopstick has 4 connection points. 2 of them are on the same circuit
trace which only goes between these two points and nowhere else (what
might this be for?). The other two, one goes to DC ground and the
other goes off into the circuitry of the radio. If I solder leads to
these four points, is that my external connection for only AM or both
AM and FM?

I presume the FM antenna is the power cord because that is commonly
done in radios of this type and adjusting the shape of the power cord
just right picks up my FM station in question, 88.7 MHz. How can I
add an external connection in this case?

TIA,

Todd K.