My method is to set the vertical radiator length where it belongs to be typically resonant on 11 meters for a 5/8 wave antenna. It is then that I tap the coil to achieve a match.
Resonance is primarily a matter of length no matter where it is mounted, so get that where it belongs before going on to the tap point. When I have used a coil of similar shape, diameter, and mounting location on a 5/8 vertical I have found the tap point to be somewhere in the 4th or 5th wrap, as has been suggested.
As for why you are seeing a low SWR at 29 Mhz I would venture that you might also see a low SWR much lower than 27 Mhz, too. My experience is that an SWR curve is not a continuous curve or else it would eventually be a circle. Instead, it is a series of curves up and down with lows and highs at different points on the frequencies along a certain number of frequencies. You usually won't see this with an SWR meter because the radio used to measure SWR with won't operate except on a finite number of frequencies. Analyzers see more. SO . . . what you are seeing at 29 Mhz is a low SWR of an antenna that is not resonant there. Well below 27 Mhz there will be another low SWR dip and the antenna will be closer to resonance there.
My advice is to shorten the antenna length somewhere in the 21.5' to 22.5' range and begin to tune with your tap on the coil. If the SWR doesn't go low enough with tap placement only, when you get really close to a good SWR it is possible to tweak the match by very slight adjustments of the radiator length. Major adjustments of the radiator length at this point is, IMO, an indication of antenna design flaws. These can be in the matching network, the radials design, or shorting issues at the wrong place between the coax conductors.
Perhaps it is a fluke, but when I have used as few as 5 - 6 wraps on a coil I needed it to be around 4.5" - 5" diameter to work well, and when it had a lesser diameter, say 3" - 3.5", it required around 9 wraps to create a good match.
Git ur dun.
