Swapping out RJ-45 connector on Differential I2C circuit
You can use RJ14/RJ25 connectors with 4P4C plug and any 2x2 UTP cable. These are smaller than RJ45.

Note, that the original board has too much empty space and too many connectors. If all you need is one I2C line adapter, you can make PCB roughly the size of 6p6c jack footprint with all SMD parts (including I2C jack) on the other side of the board.

You can use micro HDMI cable, but those get quite expensive over 6ft. Besides, cables longer than 10m are usually heavy 24AWG with standard HDMI. Most (if not all) of long micro cables are active, i.e. they have built-in signal booster chips that require power. IMHO they are not worth it, considering relatively small footprint size reduction.

UPDATE:

Just to clarify - the reason I suggested registered jack is because it is historically used for many purposes, so one cannot really guess the function by appearance. This is not so for HDMI, USB and similar connectors, where people expect them to function a certain way. If you use them for your project expect someone eventually trying to plug expensive smartphone or camera into it, and then suing you for damages.

Have you tried a simple search for "rectangular connector"? Some of them have 0.8mm pitch and footprint smaller than HDMI/USB, for example MQ172 and DF52 series from Hirose, or CLICK-Mate from Molex and many others.

Or is it that when you say "find the right cable to go with it" you mean pre-terminated cable of required length? Then you definitely out of luck, because at 20m you won't find pre-made cable for non-common connector.

UPDATE 2:

I think by fixating on minimizing PCB footprint you are missing the whole point of miniaturization, which is to minimize the space occupied by final product including all the connections required for normal operation. Let me illustrate the point: on top is "IX Industrial" connector with PCB footprint 12.1 x 9.8 mm. Note, that routing traces to those SMD pins might take up more space yet. On the bottom is CAT5 certified 4P4C jack and RJ11 plug, with PCB footprint 12.0 x 12.55 mm.



The difference for the PCB is negligible, however the plug in the first case protrudes 18 mm more from the device. If you consider minimal bending radius of 10-conductor shielded cable comparing to 4-conductor UTP you will be adding easily 40mm to your device dimensions, while saving 2.7mm on PCB

Now, I could not find ready made IX cable longer than 5m, and that one was $50. On the other hand 20m UTP + 2 plugs can be made for about $4.50, if buying cable in bulk and crimping plugs yourself.

If all this does not sound important to you, at least go with something like MQ172 series I've mentioned above. They also have huge plugs, but at least you can terminate those yourself to the cable of any length. And the jack footprint is even smaller at 10 x 8 mm.