Let's get this out of the way first............ Please keep in mind this is a 2003 truck. A short could kill your whole ride, ruin your day, weekend, or wallet. I'm just tracing wires for you. What you do with that and any consequences shall be at your sole risk. I'm not a certified mechanic, just a guy who loves a good puzzle.
AUTOMATIC DAY / NIGHT
MIRROR
DESCRIPTION
The automatic day/night mirror uses a thin layer
of electrochromic material between two pieces of conductive
glass to make up the face of the mirror.
When the mirror switch is in the On position, two
photocell sensors are used by the mirror circuitry to
monitor external light levels and adjust the reflectance
of the mirror.
OPERATION
The ambient photocell sensor is located on the forward-
facing (windshield side) of the rear view mirror
housing, and detects the ambient light levels outside
of the vehicle. The headlamp photocell sensor is
located inside the rear view mirror housing behind
the mirror glass and faces rearward, to detect the
level of the light being received at the rear window
side of the mirror. When the circuitry of the automatic
day/night mirror detects that the difference
between the two light levels is too great (the light
level received at the rear of the mirror is much
higher than that at the front of the mirror), it begins
to darken the mirror.
The automatic day/night mirror circuitry also monitors
the transmission using an input from the
backup lamp circuit. The mirror circuitry is programmed
to automatically disable its self-dimming
feature whenever it senses that the transmission
backup lamp circuit is energized.
The automatic day/night mirror is a completely
self-contained unit and cannot be repaired. If faulty
or damaged, the entire mirror assembly must be
replaced.
Some pics. They s'plain themselves to me.
Looks like the power(12 volts start/run) and ground circuits for the overhead and mirror are the same. I'd tie those two to the overhead.
Now I do believe you'll find your 20 gauge violet at the connector shown. Should light a test light or meter with the truck in reverse and no time other. Connector 201 pointed to with a purple arrow.
EDIT: looking at that connector C201 again, You want Circuit L1 which is cavity 11. 20 ga Violet with black stripe.