Insert Edge Loop
The Insert Edge Loop tool adds a new edge loop from the selected edge(s). An edge loop is a series of edges that are directly connected. They often encircle a 3D object and connect back at the origin point.
Note: Loops only continue through quads (four-sided polygons), not triangles.
Tip: You can also use this tool with the Alt/Opt+U hotkey.
This creates more geometry than Connect Edges or Connect Vertices, but it makes the geometry cleaner. For example, if you insert an edge on a single face of a cube you get a T-junction, but if you insert an edge loop instead, you get the same geometry all around the cube, provided that the loop is only passing through quads.
You can use Insert Edge Loop while you edit your geometry, and then delete the extra unnecessary edges when you're finished to optimize the geometry.