Given the radio technology of 1956 and the mountainous terrain, the small "beeper" would not have sufficient signal strength to be picked up at the distances shown.
Outside the café after Matthews tells Peterson to move, a camera is reflected in the rear fender of Naylor's car and the operator's arm is seen to move.
When the two crooks come out of the cafe, Mathews ducks down and hides behind his car. There was no reason for him to do this. He could have casually gotten into his car, looked at a map or otherwise killed time until ready to leave. Neither of the crooks knew who he was, so hiding was unnecessary.
When discussing the routes Naylor could take, Matthews says if he goes a certain direction they'd have to tail him, there was nothing else they could do. In the past and in the future, the highway patrol used a helicopter, to include keeping track of a vehicle in the desert.
The officer tells areas 3 and 5 the beeper is on channel 1, yet when he calls for the FCC he says it is on channel 2.
As Naylor is walking away from the car with a shovel, an airplane is loudly heard flying overhead at a low altitude. Since he was so paranoid over being followed, he should have looked up, but he never does.