Muonio is a small town in Finland.
North towards Kilpisjärvi
In the town center next to the intersection of E8 and route 79 towards Kittilä and Rovaniemi, there is a bridge that crosses a small stream. After the bridge along E8 there is a small road leading to the parking lot of the Kiela naturium. Hitchhiking before the small road proved successful to get a ride towards Kilpisjärvi. Also, the intersection is a central spot in the town, so people might stop just to ask, what are you doing.
There is also a bus stop after a 700m walk along the E8 that could be worth trying.
From the intersection of E8 and route 79, walk about 1km along the route 79 to get to an intersection with a bus stop. There's plenty of room for cars to stop by and also a moderate amount of traffic even on weekends.
Walk south along the E8 for about 1km until there is a petrol station next to the road. There is a small bus stop, where to hitch rides. Unfortunately traffic towards south is very rare in Muonio and it can be very hard to get rides. Very few people drive longer distances towards Tornio and Oulu, because so few people live in these parts of Finland and the nearest bigger city is Rovaniemi. That is why more often people are driving towards east than south. Therefore it is probably easier to hitchhike via Rovaniemi to get south.