@HeatherA22 I'm guessing that you have a black Hitron modem or perhaps the white Hitron modem. The model number is located on the back of the modem. It should either be a CGN3xxxx modem in the case of the black modems, or a CODA-4582U or 4582U 2A in the case of the white modem.
If thats the case, then the box in the basement will be a telephone modem, which connects to your telephone cabling.
Even before considering MoCA adapters which run over cable, or powerline which run over the house power cabling, I'd pull a wallplate off of one of the walls to check for any ethernet cabling sitting behind the wallplate that is waiting to be discovered and put to use. If that's the case, that you have ethernet cabling that can be used, it would make this much simpler to accomplish. It all depends on the age of the house.
If you had ethernet cabling in the house, they would be evident in the structured wiring cabinet in the basement, where the telephone 66 block should be located. All of the house telephone wiring would be connected to the 66 block. Its possible that the telephone wiring uses ethernet cabling. That is very common to see. If all of the house telephone cabling is connected to the 66 block, then any left over ethernet cabling that isn't connected should be available for use as house ethernet cabling. Again, it depends on the age of the house.