I find that if they KNOW you're going to call the next unit down the line, or if you're in IMC and need the service, they often will pass on your details.
If you're bimbling under VFR and not in any great need of radar service, you'll get the "freecall".
In theory (ha!) it takes the controller just as long to pass your details over his telephone as it does for you to pass them to the next controller by radio.
That doesn't always apply, of course. For those who tell their life history when calling a radar unit, or who were taught one of the longwinded mnemonics like CARPHACERIDGQ then try the short call - who are you? where are you? what do you want?