Warp Factor six is 216 times the speed of light. That means it would take roughly 1.7 days to travel one light year. To cross one parsec of 3.26 light years would therefore take you about 5.5 days. So it takes our ship 5.5 days to travel one hex one the map. Since the map is 8x10 hexes that means it would take 44 days to cross the top of the map, and 55 days to go from bottom to top! Our crew could spend a whole year of their five year mission just in this one subsector.
(generated using the Random Subsector Generator)
Now in the TV show the Enterprise went zipping around over much greater distances. That's why the authors of the first edition of the FASA Star Trek rpg didn't bother to include a map. They knew they couldn't handwave it away. But if you want to conduct your adventures on a more reasonable (and manageable) scale, the subsector maps are a good resource. So just generate four or five of these subsector maps and you'll have a sandbox that should support a five year long campaign.