For your internet modem, the location of it could affect your WiFi connections (a central location is usually best) so you might want to consider that before moving it. For wired connections, (game console in your entertainment hutch?) you could run a longer network cable from the modem to your device instead of moving the modem (may require multiple longer network cables if you have more than one device to connect).
As you suggest, if there is a path along the wall (no doorways etc. in the way) a longer coax may be one solution to moving the modem - you can join two lengths together, but use a coupler, NOT a splitter for this. Possibly the easiest. I would expect Shaw to be able to give you one for no charge or a very minimal charge. It may be faster to just go to the Shaw store if you are close instead of waiting on hold (online chat is often not too bad, either) - if you describe what you have (distances, etc) they may be able to give you a cable right there.
As to using another coax outlet, these might not all be "live" as making more than one "live" requires a splitter which cuts signal strength - so they may not have done this. Some change like moving a connection or splitter may need to be done wherever the cable comes into your place.
If you also have a box for TV connected to coax, there will already be a coax splitter somewhere for that purpose. Your best solution may depend on how this is set up. Note that your TV box and internet modem do not have to be in the same location, especially if you have a TV box that connects via WiFi instead of wire.
