Why does Miss Strangeworth mail her letters at the post office after it has closed?

Miss Strangeworth mails her letters after the post office has closed because she doesn't want anyone in town to discover her identity as the writer of those anonymous, cruel letters.
One can speculate that Miss Strangeworth isn't exactly proud of her letter-writing activities; instead, she prefers to think of this "unpleasant" task as a moral crusade of sorts. Essentially, Miss Strangeworth's sanctimonious attitude speaks of her utter contempt for those who do not subscribe to her moral views.
In the story, we are told that Miss Strangeworth likes to make her way to the post office just as darkness has begun to dim the outlines of the trees. It's far easier for her to obscure the colored letters from human view at that time.
Although there is always a group of young people gathered around the post office, Miss Strangeworth pays them little attention. The children always let Miss Strangeworth pass; as a rule, they usually stand at a respectful distance when she slips the letters into the slot at the post office door.
So, the combination of dim surroundings and the children's natural deference provides Miss Strangeworth cover for her clandestine mission.

Miss Strangeworth appears on the surface to be a harmless little old lady. In reality, however, she is a thoroughly unpleasant character, who, among other things, acts and feels like she owns the town. After all, her grandfather built the first house on Pleasant Street, a fact she often shares with tourists passing through town who stop and admire her roses. She is also an interfering old busybody, dispensing homely advice to all on her daily rounds.
After Miss Strangeworth returns home she retreats to the genteel tranquility of her pretty sitting room. Here, at her writing desk, she composes a series of vicious poison-pen letters addressed to various townsfolk. She always mails them at the new post office, "shiny with red brick and silver letters." Additionally, she always makes sure to time her visits to coincide with the onset of darkness. By doing this, she is much less likely to be exposed as the writer of all those horrid little letters. However, the best-laid plans do not always go according to plan, as Miss Strangeworth is soon to discover.