This is the default behavior of the raster calculator to use the extents of the smallest dataset. There are a few ways to work around this, but I think your quickest solution might be to first use the Copy Raster tool with your rasterized lines as the input raster and the larger layer as the Snap Raster in the "Environments" to create a raster with the same extents as the larger raster.
From here, use the Con and IsNull functions of the raster calculator. You can conditionally check areas where your larger extent rasterized lines are null/NoData (i.e. places that they don't exist) and then set these areas to the value of the already summed output(s). In areas where the rasterized lines do exist, set the value to the sum of their cells plus the already summed output(s).
This will result in an output with the same extents as "YourAlreadySummedLayers" with the values already summed for the rasterized line areas.
Putting it all together as a model might be helpful if you haven't already:
The expression in the model noted above is basically the same as my example, but with inline variable substitution: