Re: widening dirt path question
You could do it with a bulldozer, excavator, backhoe, or 4wd compact utility tractor. If soil composition and moisture patterns are a problem you might need to add appropriate remedies into the plan. If it is hard pack clay and stone probably just an outward side slope (outslope)for runoff is enough, if its wet prone then runoff cross cuts might be needed (these work nice in most cases), if really bad then an uphill side ditch with a cross culvert pipe might be best either with an inslope to it or crowned (inslope and outslope).
An excavator or backhoe can simply dig a level path (to include slope for runoff) from a lo or high starting point, either up or down. They dig and level as they go.
A dozer with 6 way tilt can VERY easily do it. I live in the Endless Mountains, steep and rocky. It's a land of logging, bluestone quarries, and recently natural gas pipelines. Dozers are the workhorse and have no issue cutting access roads across slopes. There are narrow steep skid trails put in by bulldozer all over. They work their way as they go, cut and level out in front of them.
4wd compact utility tractors are like billy goats and work fine for narrow jobs. If the operator decided it was safe enough to traverse they would drag a tilted heavy duty back blade or grader box and make an initial cut/notch and on subsequent passes make the cut deeper and wider. Once you have a notch wide enough for the uphill wheel things get a whole lot safer because you can't slide sideways with that wheel in the notch. From there a little at a time with each pass you just keep cutting it down wider.
With a 4wd tractor if it is too steep to make a straight go of it. Then you pick at it with the back blade or grader box. You pull towards yourself several feet at a time making it level continually working backwards. The engagement is several feet behind the tractor so the tractor can stay on the level and you pull 6 new feet towards you making it level. You cut in on the uphill side and deposit over the downhill side, ends up level. You can do a few hundred feet in a few hours.
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