Why might Mr. Hooper have put on the black veil in the story "The Minister's Black Veil"?

Your question correctly poses the conditional "might" as a venue to explain the potential reasons why Hooper makes the odd decision to wear a black veil in front of his face for the rest of his life. 

In reality, this is the one question in the story that never gets answered. The reactions that the veil elicits in the people of the village, however, can somewhat help us to conclude that Hooper purposely wanted to cause specific emotions among the people that reflect what lurks within their hearts. The ultimate reason why he would want this will still remain a mystery that the reader will have to deduce. 

A possible clue to help form a conclusion can be drawn from the minister's answers to his fiancée, Elizabeth, when she demands to know what is going on. Since they are engaged to be married, Elizabeth feels that she has a right to know what could be possibly driving her future husband to make a choice of this nature. However, Hooper's answers are problematic because they are not final. At one point he says to her that, if his veil were a symbol of mourning, 

"I perhaps, like most other mortals, have sorrows dark enough to be typified by a black veil." "If I hide my face for sorrow, there is cause enough . . . and if I cover it for secret sin, what mortal might not do the same?"

If he is not willing to tell the woman that he intends to marry about his rationale, why would he tell anyone else?

Therefore, we can only speculate that the minister wore the veil to cause in the parishioners every possible feeling of uneasiness. Only by understanding exactly what causes their emotions could they be able to learn the inner fears and anxieties that lead them to feel them. This would be a lesson like no other that the minister would be teaching his flock. This also "might" have been the reason behind his decision. The best evidence for this theory is found at the end of the story, when Hooper confronts those visiting his deathbed and the emotions that he conjured in them through the simple fact of wearing a veil. 

"Why do you tremble at me alone? [...] Tremble also at each other! Have men avoided me, and women shown no pity, and children screamed and fled, only for my black veil?"

Here he hits directly on how their inner fears are mirrored by the veil; hence, some equally dark and weird situations must be going on in their hearts. 

"What, but the mystery which it obscurely typifies, has made this piece of crape so awful? [...] When man does not vainly shrink from the eye of his Creator, loathsomely treasuring up the secret of his sin; then deem me a monster, for the symbol beneath which I have lived, and die! I look around me, and, lo! on every visage a Black Veil!"

With this, he directly states that he sees a veil in everyone, part of the masking and deception so typical to humans. Why fear the man who dares to expose the lies of the soul? This is a lesson he might have wanted to teach.