From Vienna south and west

If you are going to Bratislava for a day trip, thats reasonable. All of Austria is beautiful and Salzburg is a great start at enjoying it.

With 9 days in Vienna, if you get bored hop down to Budapest for one night (unless you have already been) 2.5 hours on the train for maybe 40 euro RT. Get a room in one of the riverfront hotels and hire a guide for the day to get the most out of the limited time. The only thing I would advise against is buying the return ticket in advance (and check if the hotel has a room the second night too).

Another interesting stop is Gyor and the archabbey at Pannonhalma (about an hour on the train out of Vienna.

Thanks, James E. Yes, we have also received recommendations to see Gyor and the arch abbey there. I'm also wondering if it is worth heading to the Austrian Alps (or lower Alps at least) south and west of Vienna. I think it would take 2 1/2 hours by train. Maybe I will have had my fill of Alps with Switzerland and the Hallstatt area.

You are likely referring to the Semmering rail line? I wouldn’t call the views spectacular, especially compared to where you’ve been. The Semmering rail line is notable for its engineering, not its views. I have no idea what the “cute old town” would be.

https://en.m.wikipedia.org/wiki/Semmering_railway

The mountain range south-west of Vienna is the Schneeberg (snow mountain) area, where are several hiking trails and mountain huts.
https://en.wikipedia.org/wiki/Schneeberg_(Alps)

It easily reachable by train: You take an express train (RailJet or EuroCity) from Vienna main station to Wr. Neustadt where you change to a local train to Puchberg am Schneeberg. The total ride is 1:25 hours.

In Puchberg you have the choice of walking up to the summit, i.e. for the tougher people, or usually you take the cog railway up to Austria's highest train station. This shortens the time to get to the summit considerably. The ride on the mountain train from Puchberg to Hochschneeberg is 42 minutes.
http://www.schneebergbahn.at/en/
https://webshop.niederoesterreichbahnen.at/index.php?route=product/category&lang=en&path=69

Thanks, wmt1. I think this is what I was thinking about. My question is: after Hallstatt and Switzerland, will this area be a real bonus or should I stick with places closer to and within Vienna?

The Schneeberg, having an elevation of about 6800 ft, cannot compete with the Swiss mountains, having an elevation beyond 13000 ft. The Schneeberg area appeals to Vienna's visitors because it is rather close to the city, so they can easily have a mountain experience, too.

A lot of railway fans do the ride because of the cog wheel train. At certain dates in summer, but certainly not in October, there are special trains pushed by the original cog wheel steam enginge from the 19th century.

There are no lakes like the Salzkammergut (including Hallstatt) area in the east of Austria.

Good points, wmt1! Thanks.

I am not convinced that you are thinking of the Schneeberg route as that train ride up the mountain is not really scenic as you describe. There is also no "cute old town" on the way to Schneeberg. But anyway, no mountains within the area you describe are anywhere close to what you will during the rest of your visit. Not even close.

Other day trips from Vienna that are interesting include Baden bei Wien, going up Kahlenberg, Sopron, Neusiedlersee, Perchtoldsdorf, Gumpoldskirchen.

Thanks, Emily. I have repented of any thought of going south and west to the Austrian mountains there. Your alternative suggestions are great. Thanks.