Since you have what appears to be good coax throughout your house, you might want to consider using MoCA adapters to get wired connections to the various rooms.
Here are some diagrams from GoCoax that show how to use MoCA. Be sure your splitters are 5-1675mhz to support the upper frequencies of MoCA and verify that PoE filters are installed as shown on the diagram that applied to your situation.
Might be a lot more cost effective than pulling new CAT 6 cables through the house.