Augustów is a city in northeast Poland along the E67 "Via Baltica". It is a transit point between the Baltic countries and Poland. All the transit traffic used to go pass the town, but since November 2014 a new by-pass has been opened.
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Since the new by-pass has opened, you don't need to go to the center of Augustów anymore. If you do get a ride to the city center, it might be better to get out at the on-ramp shortly before the south end of the town when you're hitchhiking towards Kaunas or at the round-about shortly after Suwałki when you're coming from there and going towards Warsaw.
You have actually three options: Your driver stops in Augustów and brings you to the spots described below. It really should be no detour for them. Option two would be your driver is passing it and goes further anyway, or worst case, the car will stop at the beginning of town. Then you have to walk for about 30-40min to cross it, following the signs for Suwałki or Bialystok, or just follow the dust and smell of the never ending TIR convoy.
At the end of town, after crossing the river bridge, there's a huge bus stop for stopping cars. Some traffic here might get off earlier, so best bet is to walk about 500m further. You'll recognize a traffic light and after this, about 50 m further, you'll find a much better hitchhiking spot. The place is also good for night hitchhiking, at least you can catch cars here towards Suwałki.
While some cars may go east from here towards Vilnius, trucks cannot use this road. No matter what your destination in Lithuania is, it is best to head for Suwałki and then the Budzisko-Kalvarija border crossing.
Walk 2km from the town centre in south-west direction along Wojska Polskiego street to 2nd roundabout along this road. Stand at the zebra crossing just after the roundabout exit on the N61 direction Warsaw. Cars and trucks are going slow enough after exiting the roundabout to stop. There is even some room for a car to pull off from the road.
South towards Bialystok
Same directions for Grajewo & Warsaw (see above), but stand on the E67 direction Białystok. There are a lot of trucks passing here from the Baltics, and because of the roundabout, they will be going slow enough to stop safely.
West towards Elk (N16)
At the first roundabout along Wojska Polskiego street in the south of the town, walk westwards. There are some bus stops on the way, so walk as much as you prefer and try to hitch cars. Best bet is to go only with traffic directly to Elk, as in between there are barely good spots to find, especially at night it can be very difficult.