You can compress it further by right-clicking the clipped tiff in the layer list and selecting Save As...
In that window, the format at the top should be kept to GTiff and set the CRS to whatever one you are using.
Go down to Create Options and change the Profile to High Compression.
If you want to do this in one step, try using the Clipper tool for rasters found at Raster > Extraction > Clipper...
Set clipping mode to Mask layer and find your mask layer in the list.
Place a check beside Crop the extent of the target dataset to the extent of the cutline.
You will have to click the Pencil button to edit the gdalwarp command to include the compression options with
-co "COMPRESS=DEFLATE" -co "PREDICTOR=2" -co "ZLEVEL=9"
Another option, mentioned by @Steven Kay is use JPEG compression (which is lossy) but works well for images that are not used for analysis, like DEMs but only for visualizing (like your scanned map!).
You can follow the same steps as above, but instead use the following
-co "COMPRESS=JPEG" -co "PHOTOMETRIC=YCBCR" -co "TILED=YES" -co "JPEG_QUALITY=75"
And optionally, consider adding overviews if you have a beefy raster that takes a long time to render when zoomed out and there are some excellent instructions at Paul Ramsey's blog http://blog.cleverelephant.ca/2015/02/geotiff-compression-for-dummies.html