How to overdrive a signal?
A 10 kohm input impedance is far too low for an electric guitar to be connected without serious loss of high frequencies. You need a buffer amplifier that has a high input impedance usually between 1 Mohm and 10 Mohm.

And I also would like to add gain control

If you add a buffer amplifier, the 10 kohm pot will serve as a proper gain control. In other words you don't need to add a gain control because there is one right there on your circuit at the left but you need a buffer amplifier between it and the guitar input. Once you have an op-amp as a buffer you can add another op-emp to perform the action of clipping/distortion/overdrive.



simulate this circuit &ndash; Schematic created using CircuitLab

I have installed 10k potentiometer between gain pins. It worked fine
on simulation but is it an effective way to control gain?

Your circuit has a gain control already; you don't need to add another.

How could I add an extra overdrive channel with overdrive control?

I've shown where it should go and there are many op-amp circuits on the web that can be used such as this one: -

Image from here.
Input buffer with 1.5 Mohm input impedance: -

The one above shows a JFET buffer amp and it's connected to an LM386 AND (incidentally) it shows an alternative way to wire a pot to the LM386 (like your original question). Image from here.
There are plenty of on-line resources if you choose to look such as this one with tone controls and buffer amp: -

Image from this site (with full parts list).