I also have a 706 and hamstick antennas, and had a lot of trouble with this setup initally. However, I made a few simple adjustments and have had great success on all the bands I've tried (40, 20, 17, and 15). I have, but don't use a commercial antenna tuner, but I am matching the antenna (see below).
Here's my not so secret guidelines for a good mobile installation. Note that I've tried this approach in my Jeep Cherokee and my wife's Dodge Caravan and it works great.
1. Power - good clean power with a good ground and short leads as possible are important. Best bet is to go straight to the battery. I have used a 20 amp aux. power outlet in the Jeep and Caravan but I don't recommend this. For example, in the Caravan there is a power outlet in the rear, but it lets RF into the DC input.
2. RF ground - you need a good RF ground at the antenna mount. I use a hatchback style mount with screws that hold the mount in place. These screws reach metal on the rear door. These doors are grounded to the chassis on both vehicles (I checked).
3. Antenna match - Initially I could not get a decent match no matter how I adjusted the Hamstick. A resonant Hamstick on 40 or 20 meters just doesn't present a decent match to 50 ohms and the 706 reduces power with even a slight mismatch. In the 18th edition of the ARRL Antenna Book I found the solution. Chapter 16, page 14 describes a cheap mobile impedance-matching coil, referred to as the "dollar special". The key component is a matching coil made from 20 turns of 10ga (I used 12 ga) copper wire 1 1/4 in. dia., about 5 in. long. The coil is connected to the Hamstick on one end and ground on the other. The coax shield is connected to ground and the center conductor is attached to an alligator clip. The clip attaches to the coil at tap points determined by trial and error for each band. In the Antenna Book there is a mounting assembly but I just left the coil in the open supported by the leads. My first contact after the coil was installed was a VE7 who gave me 10 over 9 to my first CQ. Since then I've worked a lot of DX including Australia and Japan.
So you may want to forget about the tuner, build or buy a matching coil, make sure you've got a good RF ground and good power, then go enjoy.