Do people really have a right to a phone call?
Well, I’m no lawyer, nor a cop, but hopefully one’ll be along for a definitive answer. I -have- spent a little time in the hoosegow, so I’ll take a stab at it.
Not exactly. You’re legally entitled to reasonable access to a telephone, though. Generally, these are pay phones, for which you have no change since they took your stuff. So you probably have to call collect (or charge it to your home telephone via the operator, if that’s still possible), unless you have a phone card number memorized.
Some jails’ll let you make local calls from the desk, and some maintain phones for local prisoner calls, but there’s no guarantee.
You will most likely not get to use the telephone while intoxicated (legally intoxicated, not necessarily actually intoxicated). If you’re in the tank overnight, no phone for you. You might sweet-talk a guard or trustee into calling for you, though.
So, yeah, you’ll get to use the phone eventually, if you’re in for any extended period of time. You’re not limited to one call, but your phone time will almost certainly be restricted.
Before the advent of the telephone, you got to write one letter… 
Really? I can’t imagine being able to convey much with a single letter. Maybe if you pre-arranged a set of codes with a known friend, so that ‘r’ might mean “I’m in Cellblock G, 3rd window to the left, break me out at 10pm” and ‘e’ might mean “bake a file in the carrot cake” or “now is the time to plant your potatos”
With 26 to choose among, which one do you recommend?