You’ll probably want to use a combination of tools to select the edges. First, you can convert a polygon selection into an edge selection by Ctrl + Clicking on the edge mode icon.
Problem with that is you’ll have the border edges and not just the internal ones selected, so you’ll have to deselect the borders. You can use the phong break selection tool to help with both.
The path selection tool can be very helpful for picking edge loops where the actual edge loop tool doesn’t work. Set it to live path mode and drag to create your selection (or hold down Ctrl to deselect). You can select a border of edges and then use the fill selection tool to pick the polys inside.
Unfortunately there’s no one-button solution.