Thoughts:
After the initial callup, you should not need to look up any frequencies and certainly should not have to ask your FF controller for one. When a controller ships you, he will say something like "Bugsmasher 1234 contact Chicago Center on 124.2" Once in a while, VFR or IFR, a controller has forgotten to give me a repeater change (""Bugsmasher 1234 change to my frequency 124.2"). Here you can either ask someone for a relay on your current frequency or look up a new frequency to try.
As OldPilot suggests, airport info is a good place to look. Pick a non-towered airport, though, or you may get a TRACON frequency and have to ask them for Center. Almost as easy, pull up a low altitude chart. The castellated box nearest to your current position shows the frequency that ATC wants us bugsmashers to use. Either way, no need for paper. ARTCC jurisdiction boundaries are also shown.
I like the NRST pages in the Garmin navigators too, but the numbers you see there are not a silver bullet either. For a given lat/long they list all of the pertinent ARTCC sector frequencies. So when you pick one for a callup you may get a controller for the high altitude sector that overlays your location, again necessitating a frequency change to someone else. Sectors are staffed according to workload, so there is no way to know which frequency is right for the current sector configuration. That's why Garmin has to list all of them. (IIRC the max number of aircraft that a single controller can work is 16. Sector configurations can change several times in a day.)
So you already have a couple of non-paper ways to skin the cat. The castellated box and the airport information lookup will both probably yield a better result than getting a NRST ARTCC list.