He was on ice for 70 years... but what if he wasn't?
Steve Rogers (in case you forgot) crashed a plane into the Arctic in order to save American cities from the missiles fired by Hydra during WWII. Which means that between 1942 and 2011 he missed a LOT. The world changed without him. But what if he'd been here?
Steve grew up sick and poor in an immigrant neighborhood in the 1930s and 40s. While a lot of us think about that period as being hyper-conservative, those values didn't really come in as strong until the 1950s. There's a good chance Steve Rogers would have been socially progressive. What if Captain America had spoken out against the blacklisting, the censorship, and the pressure to conform that characterized these years? Would he have been dismissed? Or would his status as a veteran and a hero have changed the outcome of this period?
The Civil Rights Era & Vietnam
I'm not sure how many people caught the significance of this in The First Avenger, but Steve was leading an integrated team during WWII- which was almost unheard of (most units were completely segregated). I believe he would have been right there with the Civil Rights Activists as the movement swept the nation. Because he never liked bullies, it doesn't matter where they're from. And I think the antiwar protests would have been significant for him as well- seeing American students gunned down at Kent State and Jackson State would have been unthinkable for him, and he wouldn't stand by while his government suppressed people's right to free speech and protest.
So postwar, I think Steve and Peggy would settle down together in New York. I think he'd be up the wall frustrated with the way Peggy's coworkers treated her- but at the same time he'd respect her decision to deal with it in her own way. So when S.H.I.E.L.D. begins I think he'd be one of the first to sign up. Would he stay? Or would Project Paperclip mean he'd turn away and find something else? Either way, I think Hydra would have been very wary of infiltrating S.H.I.E.L.D. with Cap there.