You install and configure the new switch while the current one is still in place and doing its job. In fact, you don't even need to get rid of the current switch, you can uplink the new switch into the current switch. The existing APs don't even need to be touched, they can stay as is. If you do want to get rid of the current switch, you install the new one next to the current one, then once the new one is up and running, you move each cable over one at a time. The only downtime is if the cable is connected to an AP and the AP gets the power from the switch, then the AP reboots, which is no different than if the power blips. Move one AP over and wait until it is back online before you move the next one. If the wireless network was setup correctly, one going down shouldn't affect the network, there should be some overlap by neighboring APs. If it wasn't setup correctly, install new APs on the new switch and make sure those new APs cover the coverage gaps so when you move the old ones over, them going down for 30 seconds doesn't affect coverage.
Most wiring closets already have more than 1 switch. To me it sounds like your network is in shambles and it really should be addressed properly because it could affect patient's lives.