I have a network switch, a Linux server, and my computer. The computer and server are both connected to the switch. If I connect the serial port of the server to the switch through a Serial to Ethernet adapter, will my computer be able to access that Serial Console, or will I have to connect the server directly to the computer?
NOTE: the switch is a smart managed Ethernet switch, not a Serial COM switch.
I am not trying to connect to the Serial of the switch. I want to connect to the Serial of the server through the switch.
It depends on what device your "serial-to-ethernet adapter" is. Some are just media converters: they let you run RS-232 over an ethernet cable, so you'd need a second adapter for your computer's serial port, and it wouldn't work with a switch in the middle.
If you want TCP-to-serial with your switch in the middle, you need a console server, some vendors also call them terminal servers. Some vendors will also refer to these as "serial to ethernet adapters", but that's a little misleading so you'd need to read the details of the product.