With warp drive, you jump out of warp and back into full speed very quickly, it doesn't take any measurable time to accelerate (300millseconds (last outpost). So, executing this dangerous procedure for an added 3 seconds seems overtly risky for the small amount of advantage it gives. Why would anybody use near-warp transport, considering how risky it is, to gain a minimal 3 seconds?
To save time in critical situations
Memory Alpha explains that near-warp was used during TNG: The Schizoid Man:
This procedure was used by the crew of the USS Enterprise-D in 2365 when transporting an away team down to Gravesworld to respond to a distress call from that planet. Due to a concurrent medical emergency on the USS Constantinople, Commander Riker suggested using near-warp transport in order to minimize the amount of time it would take the Enterprise to reach the Constantinople.
This emergency was a very pressing one:
While en route, a distress call is picked up from the USS Constantinople, which has suffered an outer hull breach, and is carrying 2,012 colonists.
so any delays in responding to the emergency would be quite unacceptable.
Now, it's important to remember that ordinarily, when entering a star system, a starship leaves warp and goes to impulse, the reasons for this are outlined in this question. Now, impulse is considerably slower than warp drive - this answer explains that there is a maximum of 0.92c, although realistically a speed 0.25c is more likely to be the maximum speed used. In a situation such as the emergency in The Schizoid Man, going to Gravesworld using ordinary protocols regarding warp in star systems would be an unnacceptable delay to the situation, but by using near-warp transport, it means that they do save a considerable amount of time, hence why they risk using it!