I find the two phrases of your example fit together clumsily. Pardon me, please, while I think through my thought.

“In the remaining days of the month” implies a fixed number of days. That is, today, whenever the statement was first made, through the last day of the month. I would take a trip “in” that time span, on one of the days remaining in this month.

And I would have used “this month”, too. The reason is that “remaining” implies that the number of days in the group you mention – the remaining day of the month – is fixed, is known, is certain, as of the time the statement is made. This means that you are referring to this month, the current month, even though you don’t state that explicitly. I don’t feel terribly strongly about this point – “this month” or “the month” would be understood, but I prefer “this month”.

“I usually go visit my family” is a general point of information. “Usually” implies that the action, visiting my family, is repeated according to some pattern. Maybe weekly, maybe monthly, maybe daily – but at some customary interval.

Stating a specific time interval, the remaining days in this month, seems at odds with the repeating action of visiting my family.

The customary, or usual, part of action might be to visit only in the last few days of each month. “In the remaining days of the month” implies that there are fewer days remaining than have already passed in this month. But there is no sense that the time part of the sentence is repeated for each visit, or contains repeated visits.

The numbers of actions and time intervals when they happen don’t agree.

The “In the remaining days of the month” implies that the action of the sentence, the visit, occurs exactly once during the time from now to the end of this month. The ‘usually’ qualifier on the action, though, implies repetition, multiple visits.

“I usually visit my family in the last part of the month.”

There. My soul is content. (I read that somewhere, and like the phrase.)