Warp 1+ might be anything up to warp 5-, really.
The freighters of ENT are limited to warp two, but Starfleet vessels might do much better. We never hear it explicitly said that warp 2 would be a limit. It's just a milestone in the development of Henry Archer's warp five engine - but it may have previously been a milestone in the development of Bobby McDillan's warp four engine, and before that a milestone in the development of Lucienne Ferrero's warp three engine, each of these programs involving a testbed that had to reach warp two before it reached the higher speeds.
Whether warp 5- would be enough for meaningful interstellar exploration, well, probably not - Archer's engine is said to represent a breakthrough in exploration, after all. But warp 5- would get the Valiant out far enough to encounter those exotic phenomena that whisk ships to the other end of the galaxy, without Earth immediately learning the full truth of this incredible journey.
In any case, Earth apparently had warp 9+ engines in the 2060s already. Perhaps they just didn't work well with live crew - but they were good enough to propel an unmanned probe across the galaxy in mere decades, as per VOY "Friendship One".
How far the Valiant actually got probably had absolutely nothing to do with her intended mission or her built-in performance. Kirk was adamant that the Valiant being where she was should have been "impossible".
Timo Saloniemi
, but TOS had contradictions aplenty within itself and I'm always pleasantly surprised when Berman & Co. did manage to get something right. 
