In Austria, there is something called an Einfach-Raus ticket, which is good for a full day of travel on Regional and Regional Express trains costing 33 EUR for 2 people. It can be used after 9 am weekdays, but all day on weekends and holidays.
So going to Hallstatt for a day from Salzburg would be possible, especially on a weekend. You would have to use the REX train and not the IC train, it would be only a little slower. During the week, you could not leave until the 10:39 train getting you there at 13:24. Returning on the 18:32 train would only give you 5 hours in Hallstatt. But you could leave earlier by buying individual tickets to Steindorf bei Strasswalchen, and leave on the 8:39 REX train. The train arrives at Steindorf at 9:11, so then the Einfach-Raus ticket would be valid. That would get you to Hallstatt at 11:24.
I don't think this ticket would work to Innsbruck, as for one, it cuts through Germany on the way, and two, regional trains are much slower going to Innsbruck then the Railjet train.
So yes, you could do stays in Just Salzburg and Budapest, with day trips to other locations. However, while freeing you from your luggage for these trips, you will spend more for transport costs and your own travel time. If you can travel light, one rolling suitcase each, 55 X 40 X 23 cm, plus a small day pack, it will make traveling easier.
If you want to throw Prague into the mix, I'd go from Hallstatt to Prague then to Budapest.
The more I think about this trip, Innsbruck is the outlier. If you want to visit the Alps, it is much faster and cheaper to visit the mountains south of Salzburg. Think about Zell am See, Bad Gastein, These are only 1 1/2 hours from Salzburg and can be done with direct regional trains and the Einfach-Raus ticket.