Hi George,
Yes, the iPhone is fully capable of recording your position without an outside connection (either wifi or cellular coverage). For instance, click on the following link:
http://my.viewranger.com/route/search#!3...s,United States You should see my avatar near the center of the screen. Click on the avatar and you will have the option to click on the route.
(Route link here:
Old Rag Hike Mar 17, 2012 )
This will open a new tab where you can review the route, change maps (open street map, open cycle map, google, satellite, etc... The open cycle map shows elevations and trails, so I prefer it).
Much of this hike has no cell or wifi coverage what so ever, but with a map downloaded to the iPhone you do see your location on the map. This application then (once you have either cell or wifi connection) allows you to "sync" your device(s), so the track is uploaded to the ViewRanger web site (your account).
So, I'm sure you could send an accurate gps-enabled location to responders, but I have not done exactly that. This particular application is used in Europe by SAR teams (see web site for more information:
http://www.viewranger.com/en-us/about-us/search-and-rescue), so it would seem a good candidate for what you want to do.