I need to rework this - because one thing I didn't consider: What if Maura was resigned to deal with the police up and until someone drove by and offered a ride. That possibility will change my conclusions.
A. This only applies to an on-foot departure by Maura from the accident site. (Note I think her leaving by car is not impossible, but I'm not going to address [why I think it is more likely she left on foot then not] here.)
B. I won't assume anything unlikely.
C. I don't assume she was suicidal. (Not impossible, but not addressed here)
D. I assume she didn't want to sleep outside.
1. She is probably far from where she planned on staying that night.
2. Butch Atwood let her know that he was going to call the police
3. She didn't want to encounter the police.
4. The police could come from in front or behind her on the road she was on, at any time. And they were looking for her.
5. Additionally, the person that called the police, was in one of those directions.
6. The police probably weren't going to come down an unpaved road like Old Peters Road.
7. There was some risk involved that she would be caught if she went in either direction on 112. Even if she got passed Atwood, if she stayed on the street, she risked the police driving on the street and finding her.
8. Small town, even if she avoided police on the street and found accommodation, she risks post accommodation discovery. (Bell hop calls the police.)
9. Guaranteed warmth & shelter at a police station was less important to her than avoiding arrest. (She could have stayed [or went with Atwood] and presumably been arrested for DUI.)[Remember - Atwood said Maura turned down his offer to call the police, saying she called AAA, when she couldn't have.]
Note that she had committed to all the above conditions prior to having left the accident site.
So we add:
10. She could not count on anyone offering her a ride, and thus, allow her to avoid police and obtain accommodation. Therefore, we know, that she had irrevocably committed herself to walk down either 112 or Old Peters Road, probably for a long distance, in order to obtain accommodation and avoid police. And we know she took this risk prior to her leaving the accident site. Without a tangent driver, she had no guarantee of getting a ride, so if she got an offer for a ride, it was not part of her immediate plans, that is to say if there was no tangent driver.
I think the above helps to understand her frame of mind if she was alone, on foot and was not expecting a ride - and weigh her willingness to risk a walk up Old Peters Road, or on 112 in any direction.
10. If everyone is telling the truth, (Atwood didn't lie and Weeper is correct about visibility) Maura would not be able to walk by Atwood without being seen. So East is out of the question. West is out of the question, the police came from the West and Atwood and police drove around looking West. So that leaves Old Peters Road.
1. The idea of a tangent driver includes several logical problems I've already outlined in this subreddit.
2. The logical problems associated with the idea that someone else was lucky, rather then Maura taking actions that required no luck at all, is also addressed in this subreddit.
3. Of course I could be incorrect - I can only imagine in terms of likelihood, and the above conditions.
tldr;1. we can't dismiss the idea she went on foot 2. search near Old Peters
 I have no evidence that Roscoe P. Coltrain resided in the approximate area.