Well, you don't have to multihome you main host, if you don't want too, you can just make the second network routed via the first gateway or add a static route to a new router host as long as it's on the same network. If you want to add a second NIC, then your main host will make the routing decision (specifically a "connected network" in the local routing table. Honestly, 6 of 1, half dozen of the other..excluding cost.
So, the only switch you have are the on board switch ports on the linkSys? You do this pretty easy if I'm visualizing the parts correctly. Depends on how far you want to take it, but there are a couple of ways to do it. You can use the spare linkSys as the gateway for the new network and hide the new network behind your legacy network address space, then set up a port forwarding for connectivity into the new network; telnet to outside IP:23 and hit your esx host, then use it as a jump box for the new network if you're going to have multiple hosts. Nice thing there is you don't have to change anything on your legacy network to provide NATing to the internet.
Or, you could use you cisco 1711(providing it has the Ethernet WAN) and give the outside IP on your legacy network, configure conditional NATing with source static overload behind the WAN interface, then you'd be able to just configure a static route on your main PC for the new network with the next hop address of the C1711 WAN IP. No second NIC required, just using the router to route. Both networks on the C1711 would be connected when it's acting as the gateway for the new network, so no complicated routing there.
Lol, clear as mud!
Edit: I do something similar on my home network, but for a different reason. My outside firewall is an ASA 5505 and I can only support 10 hosts on the base license. So I hung a WLAN behind a single address on my LAN. This way all my WLAN hosts appear as a single IP to my ASA and I protect my LAN hosts from being accessed via WLAN connected hosts. It's like a guest WLAN only no guests are welcome. Inside the WLAN network you can battle with other WLAN hosts, but not my all important desktop pc or game computer...or my Fort Knox NAS so full of precious booty, so to speak. The whole thing exists to reduce IP use on the ASA and provide physical diversity from my LAN. I am paranoid, but I have 3 Cisco 1200 APs to provide contiguous overlapping coverage even out by the pool, pretty much a necessity for us work from home people, so it's secure, but accessible from outside the house. You break into it you'll see cell phones, Roku, blu ray player, iPad, game councils, etc.