No Space Travel To Start With

HG Wells' The Shape Of Things To Come is a future history about social transformation but without space travel although the film, Things To Come, based on a screenplay by Wells, ends with a space launch.

The first four stories in Robert Heinlein's Future History describe technological advances with orbital space travel anticipated at the end of the fourth story and a first Moon landing in the fifth.

The three Maurai stories in Poul Anderson's Maurai And Kith are a future history about post-nuclear war recovery without space travel although Anderson's sequel, Orion Shall Rise, features a space launch. Further, Anderson's There Will Be Time, places the Maurai period in a longer past and future history with later interstellar travel.

The first story in Anderson's Psychotechnic History is set during recovery from World War III although Mars has been colonized by the time of the second story.

Apart from that, future histories begin with people already in space, I think:

James Blish's Cities In Flight begins with remote control exploration of Jupiter from Jupiter V;

Blish's The Seedling Stars begins with Adapted Men on Ganymede;

Blish's Haertel Scholium begins when Adolph Haertel flies to Mars;

Larry Niven's Known Space future history series begins with the exploration of Mercury, Venus, Pluto and Mars before proceeding to technological advances and social problems on Earth;

Anderson's Technic History begins with the exploration of the Saturnian System and specifically of Iapetus in the first story, then proceeds to interstellar exploration in the second and third stories and to extra-solar influences on Earth in the fourth and fifth.

This all sounds like a single body of work.