Fifty years ago today, John Glenn's orbital space flight ushered in a new era for space travel.
Like fellow pioneers such as Yuri Gagarin, changing history here would affect the Space Race and beyond, and would be so easy to do.
Of course, these days a plot like this will bring to mind The Impossible Astronaut and especially Day of The Moon and possibly the involvement of the Silence, as well as other SFish takes on the early space Race such a Quatermass. Setting the scene in space itself, rather than around the base and the launch platform, would do a lot to change the mood and the difficulties involved.
More on the Space Race as a setting.