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From what you describe, traumatic irritation of the trachea is possible. However, the cough may have started at the same time of the crate escape by coincidence only. As a young puppy, I would also consider an infectious cough as a strong possibility.
Since he is already improving, I wouldn't be very concerned. Benadryl will not help in this case. Because of the possibility of an infectious cough, I do not even recommend a cough suppressant.
I would continue to watch him through the end of the week. If the cough does not resolve significantly, or if he appears worse - loss of appetite, depression etc, he should be seen by the vet.
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