Feldkirch is a city in Austria.
There is a bus stop just after the tunnel that emerges after the castle. The road is the 190, known locally as Schlossgraben, it is the main highway east and is a perfect spot for hitching towards Innsbruck or further. It is common for locals to see hitchers here, so they are used to stopping at this place.
West towards Liechtenstein
There is one road that leads from Feldkirch to Liechtenstein, the 191. Walking west from Feldkirch along this road, also known as Liechtensteiner Strasse, there are plenty of places for cars to pull in. However it can be difficult to get a Liechtenstein plate car to stop, it is perhaps not too common for a Liechtensteiner to need to hitch! Worst case scenario it is only about 25 minutes to Vaduz and there is a regular bus that costs EUR 4. It is possible to even walk across the border within 30 minutes from Feldkirch.
Note, it is possible to ride the buses to Liechtenstein for free if you get on at one of the rear doors, the drivers don't check tickets.