One speaker starts working only after close to max volume
 

The car is a e36 with oem 10 speaker system and an aftermarket deck. Front has 3 on each side. Low(large), Mid(medium), High(tweeter?). Rear has two on each side. Large, tweeter. I have one mid (left front) speaker that makes no sound until volume is turned up quite a bit. Once it starts working, it works in any volume. I have to do this almost every time I start the car. I've swapped speakers left and right and with junk yard ones and even put the stock deck back in. It still does the same thing and is isolated in that location only. I know the speakers are good. I've checked all the wiring and connectors. It also appears to be in good shape. The other 9 speakers work fine. Not sure what I should try/test next.

We'll if you've move the speakers around then it's most likely the source which is your amplifier. It may be mounted somewhere in the trunk of to one of the sides.
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I'd bet it's a broken wire or, less likely, a loose connection.

When you turn the volume up, you're increasing the voltage through the wire. Once the voltage gets high enough, it can jump the gap but with very high resistance, which causes the wire to heat up. The copper wire expands as it increases in temperature and the connection is restored (or the gap is reduced enough to cause a negligible drop in power, at least from an audible point of view) until the wire is allowed to cool down again (i.e. parking your car overnight).

I ran into this situation a few times on clients' vehicles. The break was always where the wire is fed into the door and on the driver's side. Likely got pinched or the frequent opening and closing caused the wire to break eventually.

Try running a new set of wires from the amplifier to the speaker in question temporarily and see if you have the same issue. Depending on the vehicle, you may not wish to run the test wire through the door jam as that can be a pain in the ass. If you don't have the issue with the test wires, you should permanently replace the existing wire. I'd recommend using a similar gauge wire but with a high strand count. A high strand count will use many small-gauge wires braided together and they tend to be more flexible and less likely to break than wires with lower strand counts.