That will work fine...just run the wireless router as an access point. Which requires a couple of extra steps...still easily done in a minute.
Unless you have a first generation PIX501 that's a bottleneck on a faster internet connection (older PIX501 were capable of about 11 megs throughput, newer generations are around 60 I think)....no need to rush out and replace that. Granted it's older and disco'd and no longer supported...they were still pretty stable units. Also we don't know if the PIX has a VPN tunnel setup to the hospital or some facility that hosts a software package for you like McKesson or MediTech or SOS or Optimus or Pharmerica or something like that...hence no rush to replace that and break access to business critical things.
Take a PC and plug it into one of the LAN ports of the Netgear.
Say the PIX has a LAN IP of 10.1.1.1, setup your Netgear wireless router to have a LAN IP of 10.1.1.254 (assuming that IP is unused). Apply settings..and log into that new IP. Now...disable DHCP on the Netgear....apply settings. Now....uplink the Netgear to your existing network..but using a LAN port on the Netgear, you will not use the WAN/Internet port. Now you're using it as an access point, not a router. Next...quickly go to the wireless settings, assign it an SSID ..and enable WPA2/AES wireless security. Make sure you change the default web admin password of the Netgear.
Sit back and enjoy!.
For business networks, we generally don't use a combo router/wireless, it's preferred to separate the duties.