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You don't have to schedule a tech visit; there is no prohibition of you doing it yourself, or having an electrician of your choosing do the work. It is SIMPLER to have Telus do the work, as they have access to the point of demarcation, where the Filter (POTS splitter) is located, and they can easily switch the wires to include the new location you wish to place the modem.
You could also get a Boost Wi-Fi extender to cover dead spots in your house.
If the filter is a dongle placed at the phone jack, yes, you should be able to connect the wires at the new jack, and move the filter, assuming the far end of the wiring is properly connected to Telus’ incoming wire. If, on the other hand, the filter is at the Telus point of demarcation, then a Telus employee would have access.
That appears to be the filter USUALLY placed in the exterior Telus Point of Demarcation, so yes, you should be able to move it, and connect it smilarly at another.
Keep notes of the current installation as a fallback!
You should only need to place the filter on the other jack, if you are actually going to use that jack. If the jack you are moving the modem from is not going to be used, then you don't need to put the filter on it.
Are you worried about them charging you for the service call? If so, then just explain to them your issue and tell them if they charge you for the service call, that you will save them the time and switch providers. In most cases, that will be enough to get them to waive that charge.