I've attached a tool that you can use to create the interior quarter-section lines as I described in my post. In the zip there's a toolbox/script together a sample sections layer, a sample point layer, and mxd. It's a fairly simple script without much error-checking, but the workflow would be the following:
1) Run the script using your point layer as input; it will produce a text file of the interior quarter-section line segments as output. There's a distance parameter that I've defaulted to 1000 that you may need to adjust - the distance units are the same as your input point FC. It should be a bit larger than a typical side of a quarter-section (2640 feet, so I set the default to 1000 because the projection of my input point file had meters as the distance unit). The point of the parameter is that it needs to be big enough to select only the four quarter-section points closest to the section center, and no more. If you have sections that are really wonky, it may not work very well..
2) To convert the coordinates in the output text file to a FC, use the "XY to Line" tool (Data management/Features) - make sure to set the projection to that of your input point FC. The x- and y- coordinate fields to use should be obvious...
3) Merge these lines into your section lines (which you said you already had), and you'll have your quarter-sections.
Let me know if you need any clarification!
Chris