Unlike your phone wiring, which is all connected together in one circuit, Ethernet cables are not connected together, and one end is usually left in the cable cabinet as @WestCoasterBC stated. Instead of moving your gateway/router, adding an Access Point may be a better option. This would leave your gateway where it currently is placed, and adds a new point of connection elsewhere into your home. This removes the need to deal with all the other connections to your current gateway.
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