SOLVED: Networking question: Attach a wifi router to a cable modem/router?

I'm only a script kiddie so kindly reply as if you're talking to a kindergartener, lol...
My friend's restaurant switched from DSL with both a secure AND an open (guest) wifi network, to cable internet (TWC now called Spectrum aka Charter Communications). Their new Technicolor TC8715D wifi cable modem/router can not create a guest network, so they installed a TWCwifi hot spot. BIG CAVEAT: Their customers may use TWCwifi free for up to one hour per month per device (and only the smartest ones might think of using a mac changer to get more).
So, to create open wifi for the guests I put a USB wifi adapter on their bookkeeper's Windows Vista pc and set it to "AP Mode", and I put a checkmark in "Disable file sharing between clients". The Windows Vista machine has all Windows updates and the user has a strong password (i don't know if that's much help against hackers but just saying...).
I don't know how many guests will be able to get their email at the same time but with just me logging on it's very fast.
NOW THE BIG QUESTION:
Can I get a Netgear wifi router working there to create an open wifi network for the guests? Because of the way all the equipment is installed and the available space, it cannot be attached to the Technicolor TC8715D directly, I have to attach a LAN cord to a switch and then attach the wifi router to the switch. ALSO: I am not going to be able to put the Technicolor in bridge mode, due to the way the POS system is hooked up.
So that you can give me better info on setting up dhcp and the gateway etc I will attach screen captures of ipconfig/all. The one on the left is what I see when I'm logged into the secure wifi of the Technicolor modem/router, and the one on the right is while I'm logged into the usb wifi adapter's open guest network. More info: the Vista machine which has the usb adapter in AP mode, is attached to the TP-Link switch.
OH, and if you need to know this, the switch is a TP-Link TL-SF1005D and the spare wifi router I have is a Netgear WGR614v7 which has a default gateway of 192.168.1.1.

 

Thanks JoeC, I think I tried connecting a router to a router before and it didn't work, something about DHCP? and I can't put the Technicolor into bridge mode.
The obvious solution is to toss TWC/Spectrum's Technicolor wifi modem/router, activate a cable modem, and attach a Netgear wifi router that CAN make an open Guest network. The reason I didn't do that is, these people can barely turn on a light switch; when my equipment needs resetting I don't want them calling me all in a panic. (As it is, if anything screws up they can phone TWC/Spectrum because it's all THEIR crap.)

 

Thank you, Leebo, I spent a couple hours reading what go0gle threw me with no success, but the tutorial you gave me really hit the nail on the head. I can't wait to try it!
Now I wonder, can I just replace the TP-Link switch with the newly set up Netgear WGR614v2? The DVR system currently gets online through the switch and the ports are forwarded to the DVR's IP on the LAN...

 

And... DONE. Just like you said, Leebo, everything's working like a charm, THANKS!
The tutorial you found was the most correct and complete info I've ever seen on this topic, but true noobs will have to go back to go0gle and study the details of most every step. For example, the tutorial mentions "ipconfig release/renew" which is not the exact correct text to type into a cmd window. Also, you have to read the comments at the bottom to get clarification on choosing a new range for the DHCP pool.
Still, it was more than good enough for my needs. Thanks again!