I'm just out of range to connect to one of the networks in my neighborhood. I'm currently running a Toshiba Satellite A75 with an Atheros AR5005GS Wireless Network Adapter. I purchased a Linksys WPC54G Notebook Adapter, but I'm not sure that it will help so I'm not taking it out of the box until I'm sure. Will installing this adapter help me? And/or can you tell me some other way to do it?

Well first of all I hope you have legit access to their network. But anyways, if you're bored build a cantenna: http://www.turnpoint.net/wireless/cantennahowto.html

And get a nic that will work with it.

well I didn't feel like looking up your cards but here's what I've got going that works.

Zcom 300mW card w/ both a 7dBi anttena and a 19 dBi flat panel anttena (sp?)

Try looking at http://www.wlanparts.com/ for other options. My setup with the flat panel can go over a couple miles really easy... it's really just line-of-sight. I wouldn't doubt if it could go over 10 miles. They also used these cards for the DefCon 170+ mile setup.