You can reach a major highway rest area without too much trouble where you should be able to find a decent lift.
Take bus 353 until you reach E 170 St and Cottage Grove Ave. I believe it starts from Millenium Station but can also be reached by taking a couple buses from the city.
When you get off head towards the railroad tracks and turn left on the very first road after the tracks. You can even cut across a field to save a few steps. You'll then come to Parkside Ave and just need to follow that until you reach the end of it. Then turn left on 173rd Pl and you will shortly reach the employee entrance to the rest area right on 294. All in all it is about a 20 minute walk from where the bus drops you off.
There are plenty of restaurants here and a gas station so lots of people stop here instead of pulling off the highway at an exit. And the services serve both sides of the highway via a bridge going across so one could use this to go East as well.