Horgos-Röszke border crossing
The crossing is situated on the E75, which is a motorway to Budapest and an express carriageway to Novi Sad. There is heavy traffic, both trucks and cars. Walking across this border is permitted, and in fact you should get out of the car and walk across because there is often a long wait for border controls.
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On Hungarian side walk for 300 meters and you will find a petrol station where you can ask drivers. Be aware that most of the cars are Serbian drivers going somewhere nearby, so it is not that easy to get away, but you will not starve there either.
Whether you can hitch from the border depends on the mood of the police that day. Sometimes there's no problem with standing a few meters down from passport control, and you might even see other hitchhikers. At other times, the police may ask you to move to the petrol station 1.5 km down the road. However, if they ask you to move, it often suffices to walk another 200 m down to the green road sign and hitch from there.
If you want to get a lift to southern Serbia, Macedonia or Bulgaria, it is best to write a sign "Bulgaria", because very many Bulgarians are driving back from Western Europe through this border crossing. Even if you are not going so far, you can rely on them to leave you at, for example, the big toll station after Beograd.