Is it possible to manually change Network position of Network Analyst Stop without moving the stop?

Network Analyst does not always choose the street I need.
When I get a new .mxd file, the Route territories have been defined and the Network Analyst Stops placed centered at the nearest network street (within the county parcel shape) or where the position has been marked from a previous visit.
When I Solve, the solution does not always take me to the correct road.
The Network Analyst Stop graphically represents the target, but sometimes the house and driveway are not on the same street. Sometimes the Property and Parcel are landlocked with no network street associated.
Without moving the stop from the house (target) to the driveway, how can I manually change the position on the network?
This is difficult in rural areas where the roads and driveways are thousands of feet apart, but also in downtown areas. A house at 101 5th Avenue will be routed around the corner to Baker Street. Since there is a sidewalk easement on the front of the parcel, the Stop is closer to the side street than to its address.