In the mid 22nd century and later, the United Earth Space Probe Agency might have been a common government bureaucratic subdivision within the United Earth government.
So you would have a United Earth
(Space Probe) Agency
, just like you would a
Archer's Starfleet would have been part of the UESPA, the MACO part of the UEDA, even after the founding of the Federation the UE and the UESPA would have continued to exist.
If United Earth initially started in the 21st century as an international organization, not a government (maybe a opposition - counterpart to the Organization of the United Nations) then it could have been in existence long prior to the 2150 date usually given for the UE becoming a state. This would have allowed it to be in a position to launch warp driven probe immediately after Cochrane's first flight in 2063.