The key to carrying hiking maps on the Via Alpina—especially when walking a section that necessitates multiple maps due to covering multiple countries and regions—is to carry only one or two maps at a time while planning when to pick up the new ones you'll need to get.
One of the most useful pieces of information that Via Alpina's Web site provides is a list of the pertinent maps you'll need on any given day. This information is always found below that day's stage description. There's also a selection of mapmakers to choose from.
I prefer the 1:50,000-scale maps because they provide enough detail, but still typically cover more than one stage—that way I won't lose my shirt buying topographical wallpaper. Some folks swear by the 1:25,000-scale variety because of the assurance of detail.
Regardless, one of the biggest plans you can make along the trail is keeping your eye out for likely places where maps will be sold. Be steadfast about planning ahead for the time when you'll replace your soon-to-be outdated map. It's a queasy feeling to start a day and know that if you get lost you're without the proper info to get back on the trail. In nearly every midsize town along the Alps, there are bookstores with the topo maps the VA site suggests. In most cases, the employees are also well-versed in the ones needed.
When your map has become outdated—i.e., you've finished those stages and survived for another day of hiking—you've got three choices as I see it:
A) Keep it and add more weight to your pack;
B) Send it home;
C) Help out a hiker in need who's likely panhandling outside an Austrian bookstore as we speak.