If you add a 2nd phone to your existing account then it will be on your existing carrier so the same dead spots.
If you go with any of the el-cheapo phones. Well some use Sprint (Avoid) some T-mobile (At one time T-mobile phones roamed on AT&T I do not know if this is still the case. My last phone or the one before it showed Verizon towers but would not connect where I was.. May have been where I was, ASK before you buy)
and some use AT&T
And some use Verizon.. Not sure which use which service but you can get service (Voice only( for about 20/month if I'm not mistaken. Just make sure it is the carrier you want.
NOTE: I am NOT a fan of Verizon but that has nothing to do with this thread.
Another possible solution.
A pair of battery operated 2-way radios. or better yet. A Technician Ham License.. Amateur Radio... When all else fails.