Unless you are taking off from an airport that is 3 miles from class C and flying directly away from it immediately, I would establish communications.
First of all, you are of course in the radar area that they control despite being outside class C. So it is probably not a difficult thing for them to create a strip for your blip on the screen.
Secondly, I can't remember what words the AIM uses but if I remember correctly it says communications are either recommended or encouraged. i.e. if the controller sounds like they have time, give them a call if you are going to be flying in the area.
Thirdly, I have a story about this that supports the first two reasons I just typed.
I had about 25 hours in my logbook. I was going up with my instructor at night from our class D airport to an uncontrolled field that involved flying through or over top of a nearby class C airport to get there. Our class D training airport's airspace actually touched the class C's outer ring and there was a cutout. So the point is, it was right there.
We departed and had told our class D tower we were going to the north. They wanted us to stay on frequency until traffic coming from the north was clear. So we did, and leveled off at an altitude that kept us below class C but we would have liked to be on the radio with the class C controller by then.
Well the traffic went by us and the class D airport tower said we could switch frequences so we flipped over to the class C controller's freq. First thing we heard:
[sound of mic being keyed with Traffic! Climb. Climb now! Climb. Climb now! in the background]
"Approach United 751 we're climbing for an RA."
We saw them about two miles ahead of us and probably 500 to 1000 feet above. We were not in class C and they were doing as they were told, yet we generated an RA since we were climbing and they were descending.
The class C controller was quite irritated when we checked on and requested VFR flight following to the north through and above his airspace. We got "the phone number" radio call.
My instructor called them after we landed and they basically told him/me, 'we know you were doing everything right and you were not in our airspace. We also understand that you would have normally switched over by then but tower wanted you to stay. In the future, we'd appreciate it if you could switch over sooner.'
Basically my point is, even though a plane could be outside class C airspace, it could still really be getting in the way of things if it's on a 15 mile final at 3000 feet, for example, orbiting around maneuvering there. So it's best to let the controller know what the plan is instead of having them have to guess as to which way this blip on their radar is going to go next.