The evidence for warp .5 is right on the screen in TMP. Kirk's orders and Captain's logs make it plain.
Kirk's first Captain's log occurs after the ship flies by Jupiter and in it he says, "...1.8 hours from launch...", during which time the ship has been traveling warp .5.
At its farthest from Earth, Jupiter is about 3,220 light seconds away, or about 54 light minutes distance. At half light speed, it's 108 minutes away. 108 minutes = 1.8 hours. Ergo, if Jupiter were its maximum distance from Earth when the Enterprise launched, at half lightspeed the ship would get there in exactly the 1.8 hours Kirk cites, which is too on-the-nose to be an accident.
But, for the sake of argument... At closest approach, Jupiter is 1972 seconds/32.87 minutes away at light speed. To over that distance in 1.8 hours at warp .5 would put the ship at .30c (just shy of 1/3rd light speed).
So, is TMP is any guide, warp .5 is somewhere between 30% and 50% of light speed, with 50% being the more likely intention.