Can Ethernet Switch Connect Directly To POE Adaptor?

I suspect you mean poweline device which run ethernet over the electrical power. POE is normally a term used for providing power to ends devices.

Poweline devices can be hooked to switches if you like. Powerline device are a form of switch when you look at the function they perform.
 

KHE920

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There is an antenna on my roof (rural internet). An ethernet cord comes from it to a POE adapter which is plugged into an outlet in my home. The POE adapter has another ethernet cord coming out going to my laptop.

I want to have more ports, so I want to connect an ethernet switch. Not sure if I can connect it directly to the POE a daptor or if it needs to connect to a router first?

Back when I had landline internet I think I tried to connect a switch to a modem and it didn't work. I think I needed to connect to a router first.
 
I see that device is called a power injector and it is pretty much transparent as far as data signals are concerned.

It depends what the device on the roof is. As you correctly surmise if the device is a modem you can only hook 1 device to it if it is a router then you can connect multiple just using a switch.

Hard to guess. If the ip it gives out are 192.168.x.x address it likely is a router but ou never really know.