Solar cell efficiency on self tabbed cells
Supplying a photo of your cells and your soldering would be a very good idea To be certain that what you are describing is properly understood, but ...

The following is based on my experience in this area:

A partial attachment of the tabbing wire to the solar cell should work properly. If you make as good a solder joint as you can, and have a good joint along say at least 10% of the connection area, then the majority of resistance to current flow will occur within the cell. 

You can test this by soldering tabs to some cells with as much contact area as possible, and soldering some other tabs with very little contact area.  Expose all cells to sunlight and measure &amp; record current flow when the cell is short-circuited with an ammeter. There will be some variation between cells, but if the soldering is OK you will not be able to tell which way the cells are soldered by looking at the short circuit currents.