If you need more gain from your INA103, remove the 10 ohm resistor in the gain loop & try replacing it with a 4 or 5 ohm resistor. Why? I think your 1k pot is not actually measuring "0" ohms at it's minimum - you probably have 2 to 4 ohms at the minimum position. That's fairly common.
Using the gain formula included in the INA103's data sheet, a true 10 ohms in the gain loop will only give you 56 db of gain. I would guess that 10 ohms, plus whatever residual resistance your particular 1K pot has, is just not providing enough gain for your needs.
A 6 ohm total resistance in that gain loop gives you 60 db of gain - that's probably enough for most applications. Also, by using the same gain formula, 1010 ohms (your 1K pot plus resistor) in the gain loop still gives about 16.5 db of gain. So "off" on your pot is not "off" on the gain setting of the device. That's why you hear a faint signal in the off position.
If you really want to be precise with the gain settings of this chip, drop the pot & use a shorting rotary switch with 1% resistors for gain setting. The 1k gain pot approach is "ballpark" at best.
If you try to increase the gain on the output of the chip - as shown in the data sheet - you will uncover one of the INA103's major shortcomings: a wimpy output stage in it's instrumentation amp architecture.
So, I hope that your next amplification stage has a very high input impedance. If not, the INA103 will have difficulty putting out enough current without some distortion. To solve this inherent problem, maybe a current booster circuit in the INA103 feedback loop would be in order. That would entail adding a suitable current booster or buffer circuit on the INA103's output, taking a parallel feed from the output of the current booster & going back into the INA103 at pin 11.
If you try to add more gain by re-configuring the 103's output stage with additional resistance in the feedback loop (as shown in the data sheet) you may get into aural trouble - especially if the next stage's input impedance is below 100K ohms.
The INA103 is a pretty cool instrumentation amp. It allows access to the feedback loop on the output stage which is something the SSM2017, SSM2019 or the INA217 do not offer. This chip just needs to be goosed on the output current & then have the gain set accordingly. Good luck!!