It depends on the jail. Our jailer will let a prisoner write down any numbers he needs before turning the phone off and sealing it with the other property, unless the prisoner is particularly obnoxious. I have even looked numbers up for prisoners (I am generally doing my paperwork in the jail while they are being booked in) that have been cooperative and easy to deal with.
Once one is arrested, his freedom and rights are restricted. That is part of being arrested. Jailers in larger cities can be overwhelmed with book-ins, or just not have the time to devote to each prisoner individually, it isn't a hotel. For safety reasons, outgoing calls are monitored. I don't let anyone call anyone once they are in custody until they get to the jail, sorry. All I need is an angry mob that can stem from one phone call.
And, if a prisoner is obnoxious, rude, fights, resists, and makes my life difficult, I am reluctant to cut him a break or be helpful in any way. Sorry!
And you would be AMAZED at the number of people who not only don't know the phone number of an emergency contact, but don't even know their last name! "Emergency contact or someone you want to call?" "I guess my boyfriend?" "Name?" "John." "John what?" "I don't know. It starts with a T."
I always at that point stop my paperwork, shake my head, and start again.