The linked question was about one particular story - hardly enough to generalise from. I will, however, try to answer as best as possible.
Odin as a patron
First, there is one very clear example of Odin being a patron of an outcast - the church thief Ragvald Odinskarl, in late 15th century Stockholm. Of course, this is much to late to be taken as definitive proof that Odin was originally a god of outcasts - Ragvald could probably just as well have latched on to any other of the old gods.
If we turn to a time and place where we have - fictionalised - accounts of outcasts, and some evidence of patron gods, i.e. saga-age Iceland, we do not find any example of any of the outcasts having any particular relationship with Odin - he is actually hardly mentioned at all outside of poetry. Thor seems to have been a much bigger deal, and for Frey there is Hrafnkels saga Freysgoða. One would think that as the sagas were often about poets, something with aspirations to be chieftains on Iceland, Odin would have been a decent choice of a patron. The sagas were, however, given final form by Christian writers, so it is possible that most passages painting the old belief in a flattering or even neutral light were exercised unless integral to the story. If the saga heroes does turn to someone for supernatural aid, it is usually a Christian figure.
Odin as an outcast
If we turn to the mythology, there are a few aspects of Odin that does mark him as not totally respectable, even by Viking age standards. First and foremost - and this is how Loki attacks him in Lokasenna and thus something likely seen as very important - is that he practices seiðr, which was seen as very unmanly, and according to Loki even went about dressed as a seeress and prophesied. The habit of wearing disguises, often turning up as a poor traveller, is something that comes up in many stories. Odin is a trickster, who prefers to take what he wants through guile, and seems to like to fool others even when doing so does not seem to confer any actual advantage.
Finally, he is the god of the hanged. This is, however, not linked to justice, but rather to sacrifice and death in general, and the seeking of knowledge from the dead. Odin does not offer comfort or aid to those about to be hanged, he is interested in learning what they know.
The first part of the answer is mostly based on original sources or easily found knowledge. I confirmed that Odin is not mentioned often in the sagas of the Icelanders through this search of sagadb.org. A few sagas might be missed, but the important ones all seem to be included.
The second part, and in particular the discussion of Odin as god of the hanged, is based on Gro Steinsland's treatment in Fornnordisk religion.