You are doing it the right way, one month at a time.
My strategy was to write down each person I wanted to search for and their location and their timeframe in each location.
Then it is just a case of working down the list, ticking off as we go.
Now I tend to take screenshots of the articles and save them i.e. name them in a format that suits SmithJohn b1880-01, 02, 03, SmithJohn b1910-01, 02 etc
If you see a deal with Find My Past it can also help as they also have Newspapers. So it can be worth using both. That said I prefer BNA as the FindMyPast version does not have good filtering.
I have a folder system for storing data and the Newspaper Screenshots would go in folder Genealogy/05 Newspapers and Periodicals