I'm thinking a detached garage based on the picture or if the similar house is like yours with the back yard that low, 1/2 way through the depth of the house maybe you could do a lower level garage where a bunch of the structure ends up hidden by grade.
Maybe a slightly further shot of your actual house might help with better site ideas.
I think at first glance most would agree with your initial fears of looking ugly.
Do you have budget to do a house expansion or only garage?
If you built another whole section like section of existing garage you could maybe get 3 in front and 1 in the side at the back (where the windows are). Then you would get either shop space in existing garage area or living space. Plus additional upper living space but you would loose the windows in the room behind the garage, assuming similar is accurate.
Existing floor plan, site topography, side building restrictions all play a critical role. A good designer can help with some of the exterior lines to tie it all in to not look like a hacked add-on. Other than that it's just playing with what you need/want and how you can achieve it by adding on or building close by.
A detached garage with a coach house on top might be simplest way to compliment house design and even be a potential revenue stream but buildable area may take that right off the board. Here coach houses are a hugely sought after thing either by renters or potential landlords. If you can handle being a landlord, even if you farm out the management tasks to a professional, it could be a no brainer if there is any sort of rental market in your area assuming local rule writers have made allowances for such.