I have the cat5e throughout the house but in a ring topology as it was wired for phones only. RIght now I just have two points connected end to end. Next step is to think of a clever way to use the 2 unused pairs to support another 100BaseT and create a fake hub and spoke topology. This G.hn standard might be a good way to get even more performance out of this setup.
[0] https://en.wikipedia.org/wiki/HomePlug
(edit: just noticed "ethernet over power" mentioned in another comment here. Same thing)
The cable line in my livingroom connects to a modem. The modem feeds into a wifi router so that I get a good signal throughout most of the house. One of the ports in the router's on-board switch feeds into a wall outlet, which connects to a master switch in the basement. I could never go back to pure wifi.
What I find strange is that, like the OP said, the original network was setup for phone outlets... but the house was only a couple years old. Are RJ11 phone networks really still a bigger selling point than RJ45 switched networks? With the high adoption of cellphones and the increasing demand for internet-connected devices, it seems like RJ45 should be standard.
I'm out the cost of materials (maybe $200?) and the price of having an electrician add an outlet to my "MDF"/hall closet ($175).
Where a cable needs to make a turn to follow a corner, I screw in a cup hook. For support along a straight section of wall I nail a wire nail partly in and hang the cable on the protruding part.
https://i.pinimg.com/736x/43/cc/e2/43cce2542a8cd5c3ca13f3604...
Staples leave small holes when removed which are easy to patch. You can also use many of them to get nice straight and tight runs. You can put a bit of white (or whatever color your walls are) paint on them to help blend in.
It's not ideal aesthetically, but nice straight lines and right angles go a long way to making it look better. The next step up would be to run some channel to hide the wires, but I don't find that necessary.
If you put your modem at the same jack where the first tee in the line is, you should be able to get at least two connections, but for more than that you may need a switch at each tee.