According to me “par hasard” is a politeness. When you are asking something for example : “Avez-vous un briquet, par hasard?”. It makes the question less direct, facilitates a negative response by suggesting a positive or negative response.

But in your example, I don’t know why, it sounds kind of ironic to me. Maybe it is suggesting that you spotted that the person who is avoiding you thinks you will not notice it. You are implying that you are not dumb.

“Par hasard” could be here the equivalent of “is there any chance that”, so I would say that this is somehow making the sentence smoother and thus increasing politeness.

An old Tintin springs to mind:
“Tu te moquerais pas de moi, par hasard?” As the speaker shortly after stuck a swordstick through the other man, I do not think it was meant politely, in that case at least.

It really depends on your intention.

As is, adding “par hasard” in a question asserts that you sincerely ignore whether your interlocutor can satisfy your request, that you are not asking him something with the purpose of letting him no way to say no.

Auriez-vous l’heure par hasard ?

It would be technically useless to add “par hasard” if you saw your interlucutor giving the time few seconds ago.
Saying “par hasard” asks if there is a coincidence in your favor, saying “s’il vous plaît” just asks if your interlocutor is willing to help you if he can.

But, as @Pogus says, it can be completely ironic too.
Taking the same example above, if you ask the time to somebody that just gave it and saw you watching him giving the time, he would then know that you are not asking “par hasard”, you are asking him on purpose, with the conviction that he can fulfill your request. It feels like a trap.

So i would say it can be a indirect politeness by default but its sense is related to context.
If you want to be polite with no ambiguity, simply add “S’il vous plaît” imho.