I too am a little confused over the OP's exact intentions, so here's some info that may or not be relevant to the question at hand...
1. Verizon's Mobile Hotspot (MH) app. Verizon has their builtin MH app that can be used to "tether" the phone's data connection to a computer or other device via wifi. *IF* you have a tiered data plan (i.e. not unlimited data) with Verizon, you can use this app free of charge. If you have a grandfathered unlimited data plan, VZW charges something to the tune of $30/month for access.
2. Other hotspot/tethering apps. There are quite a few other tethering apps out there. Some are wifi hotspots, some require the phone to be connected to the computer via USB. Some require root access to work, some don't. Some work very well on the S3, some do not or only work on certain software versions. Personally, I don't have the experience to be able to actually recommend one hotspot/tether app over another, but I'm sure there are others here who can.
The obvious caveat to either method is that any data you use over a tethered connection does, in most cases, count against you data plan -- so that's something to watch out for. Also, using a tether app used to be against Verizon's TOS (unless you subscribed to their MH service), however, I have read that current FCC regulations regarding the LTE spectrum Verizon uses prohibits them from preventing users from using tether apps (but they are still allowed to charge you to use their app if you choose to do so). This is just what I've read...I could be 100% wrong about this and do not claim to be an expert on this subject.
In ALL cases, we are talking about using your phone as a gateway to connect to the Internet on other devices via the phone's MOBILE DATA. These are alternatives to wifi when a connection is not available. These will not improve an existing wifi connection nor extend it. In almost all cases of using tether apps myself, you must have wifi turned OFF on your phone to even activate it...so...yes, you do at the very least have to have some kind of data plan on your phone for this to even be relevant.