Had to think about this dilemma which is common in snow country.

Being 150 miles north of you - and getting in excess of 127" of snow this season - we luckily had consistently cold enough temps so there was little to no daytime melting and refreezing. In the spring, the huge snow piles just evaporated without much puddling and running water.

But I've experienced it before and it can be solved without building a new garage... if you have the Minnesotan's required long handled "ice chopper" (they're a common implement and sold everywhere from Menard's to Walmart) and are willing to do a bit of work when the sun gets low, temps are dropping, and any new melt is starting to freeze over.

I don't know how high your threshold is underneath the garage door, but make certain any new accumulated snow is cleared away. Chop back the forward softer or crackly edge of the ice and throw the chunks on the lawn - the next day's new water should then hit a fairly solid ice wall. Keep working at chopping the front edge back.

Then work on chunking out a channel somewhere between the forward ice edge and garage threshold. That's what takes the diligence, but it will direct water to a low spot on the outside of the garage - which I assume is the right side by the chain link gate in your picture.

If you start this early and keep at it, you should be okay.

The only time I had a real problem was a few years ago when the floor drain and pipe to the outside froze; when we did get a melt, it overtook my threshold and I had a skating rink for a while until the pipe thawed. That's never happened before or since.