Ethernet switch only works when immediately plugged in

Here are the laundry list of items I would try:

1.First test is always to eliminate the switch and see what happens.  Connect as many of the devices which are hanging off of the switch directly to the router or through another switch and observe what happens.   If you still see some version of the issue, your problem is probably with the clients.  

2. Not much an unmanaged switch can do to prevent the devices from communicating.  Next question is if you notice any differences in the activity indicators on the switch between when it is first powered up when the devices are accessible and after they disappear.  If you are no longer seeing the activity indicators blinking, then the issue is your switch...needs to be replaced

3. If there is still activity then the issue is most probably something to do with the clients.  When you reset the switch, get the IP addresses of the items off of the switch and when they seem to no longer be connected (but you still have activity lights on the switch) try pinging the IP addresses to see if they are still at the same IP address and responsive.   Another suspect item when traffic mysteriously stops shortly after you put something on to the network is that you have a duplicate IP address.   Look at the items that you are connecting to the network with the switch and make sure none of them have static IP addresses, or that the IP address they pick up is not the same as another on your remaining network.