New wall Ethernet jack needed

If it is 4 pair CAT 3 which I assume it is it technically can do speeds 1Gbps or more at short distances. You would need to change the telephone jack to an Ethernet jack then trace another wire in another room where you want it back to where this wire meets and bean it then also install an Ethernet jack on it on the other end as well... Typically this would cost $200 for the appointment + $75 to activate a jack... Or $200 + materials and labor cost to run a wire... Not typically something done in a condo as it would be done externally on the wall... You'd likely want a low voltage technician to do this if you want it hidden... Probably would require some drywall repair and painting as well going that route... Personally I'd just use the old cat3 converted to Ethernet and call it a day. Good luck. If you have the technical nohow I'm sure you could technically splice it yourself but you'd need the proper tools and know how to trace wires etc... likely the cat 3 lines all meet at a jack in your closet.

You might be able to do it yourself at minimal cost depending on how far you have to move it and what is between the two locations. Ethernet cable is cheap and easily available.

Should be a simple job for an electrician to do in under an hour depending on your layout.