I'm a little unclear as to your situation. If I read it correctly, the derailleur has trouble moving down the cluster, especially 7 to 6 and 3 to 2. What's confusing is you mention inadequate cable tension when there's a problem going from 7 to 6. When the shift is made going from larger cogs to smaller (lower to higher gears), the shifter is releasing a set amount of cable and overall cable tension is decreasing. The symptom of over shifting from 3 to 1 may be a result of not enough cable tension to prevent the derailleur from moving all the way to the start position under the small cog.
At any rate, be aware that the number one cause of shift problems is excess friction somewhere in the system. This could be caused by excess dirt or a physical flaw in the cabling. First thing to do is check that your rear wheel is fully into and centered in the rear dropouts. Examine your cabling including the ferrules (outer cabling caps) for wear or contamination. You could disconnect your internal cable from the rear derailleur, and while holding it with your fingers, manipulate the shifter through its range. Every shift should cause the inner cable to move smoothly without any areas of resistance. If you feel resistance, you could isolate where it's happening by moving you grasp up the cabling to the next area of exposed inner cable and shift again.
If it appears to be a cable issue, it's best to change the entire cabling, both inner and outer as well as ferrules. Damage or wear in one area infers that another spot may be close to worn as well, and, besides, you can't just change a section of inner cable.
Another thing to check is if the derailleur hanger is straight. This is a little complicated involving a specialty tool. However, a good estimation of the hanger straightness can be seen when looking at the hanger and derailleur from behind the bike. The hanger should be straight down and the derailleur cage parallel to the rear cogs. A bent hanger can often be replaced or using the hanger alignment tool to correctly align the hanger