In TOS, human settlement ranges across an area several thousand light years across and perhaps a thousand light years in thickness. It is unlikely that any two acknowledged human settlements lie more than 4,000 light years apart; in TOS, this is perhaps four months' easy cruise for a starship, and a starship located around Sadr should be able to reach any spot in the UFP within two days at emergency speed. Extensive charts cover everything the size of a planetary body or below, at least to within a thousand light years of Earth on the galactic plane.
It is important to note that this is not continuous, and - although all major bodies have been charted - not necessarily explored. Due in part to a policy of noninterference, a number of independent civilizations exist within the area as well, and only a small fraction of the systems are settled.
During the first two seasons of TOS, frequent references are made to known stars. Rigel appears to be one of the central "hubs" of the Federation; "Mudd's Women," "The Menagerie," "Shore Leave," "The Doomsday Machine," "Wolf in the Fold" and "Journey to Babel" all independently reference various planets in the [large] Rigel system, travel to and from Rigel, etc.
"Wolf in the Fold" establishes that human colonies go beyond Deneb. Although the exact distance to Deneb is not know very precisely, Rigel is somewhere between 700 and 900 light years away, and Deneb 2-3.5 times as much.
When Zefram Cochrane asks after civilization in "Metamorphosis," Kirk tells him that we [humans] are on a "thousand worlds." As we see in various episodes, however, many colonies in the TOS era have a very low population.
In "Squire of Gothos," the crew is quite surprised to find a free flying planet 900 light years from Earth while en route to a Federation colony. What is particularly surprising is that an object of that size is not on their map; thus, charts include even small bodies.