I'm so sorry to hear that your pup is hurting right now. First off, don't take any of the precautions you would for pain in a human; pain killers made for people are hard on the stomach of a dog, so if you run out of the painkiller prescribed by the vet and the problem persists, be sure to go back for another prescription. Your vet should have already checked for all the obvious factors. Most of the time limping is because there is pain in the dog's joints; but the cause of that can be difficult to find. If the vet is certain there's no damage to the leg he was limping on, there's probably some minor discomfort the puppy is going through depending on how long he has been limping for. The pain killer should generally be enough to get him through it until he heals if there's not serious issue. He's a young dog so he is unlikely to have arthritis or more serious issues. If he's been limping for a long period of time then there's a higher chance something else is going on.
If you want to reexamine him yourself, start off by checking to be sure no areas have a particularly painful reaction. Do not be too hard on the dog, and if he starts to seem panicked or becomes aggressive, stop for the time being. Make sure to check his toes, including the toenails for any cracks because this can be very painful for a dog, the webbing (the area between the toes which could have been damaged while walking if he stepped on glass or some other sharp object), and the joints. A dog won't always yelp or pull away when they experience something painful while being examined this way, so the best sign that there is something wrong to look for is if he raises his head at one action specifically.
If there's no sign of what could be going on and the limping persists for longer, return to the vet and insist on a more thorough exam. If you're able to pinpoint what the problem area is, let the vets know what it is.
First off, don't take any of the precautions you would for pain in a human; pain killers made for people are hard on the stomach of a dog, so if you run out of the painkiller prescribed by the vet and the problem persists, be sure to go back for another prescription. Your vet should have already checked for all the obvious factors. Most of the time limping is because there is pain in the dog's joints; but the cause of that can be difficult to find. If the vet is certain there's no damage to the leg he was limping on, there's probably some minor discomfort the puppy is going through depending on how long he has been limping for. The pain killer should generally be enough to get him through it until he heals if there's not serious issue. He's a young dog so he is unlikely to have arthritis or more serious issues. If he's been limping for a long period of time then there's a higher chance something else is going on. 
