I spent a long time making street atlases! I worked for a few companies. They all had their own way of doing stuff. It used to be mostly CAD files that were built in a tile format from USGS Quad maps as a base and then expanded on. At Mapsco we used tiled USGS 24,000 Quads in CAD format made up the base and were imported into AutoCAD Map. We used templates for drawing everything in CAD. I dont think there are any really firm rules on how to design a street atlas, especially since its been so long since they were really in the public consciousness.
Right now where I work (County government) we use data driven pages in ArcGIS and a grid file to export each page of a street and cadastral atlas for our employees to use. You can specify an overlap and annotation would ensure repeating labels where you want them. It exports everything to PDF as a single multipage file.
If I were going to make a street atlas that looked good and wanted to keep updated I would probably make the entire base in one file in Illustrator, and then you can split up the base with various artboards and export it.