The rawhide is now in the stomach. How much had it been chewed on? Was it soft and pliable when he swallowed it? Since the rawhide is in the stomach making him vomit to try to get it out is an option. If it is successful and they vomit it up, then the problem is solved, however there are a number of risks to this approach. First, there is the possibility that it could get stuck in the esophagus on the way back up and cause him to choke or just lodge somewhere that is very difficult to access to remove it. Second, he could aspirate and develop pneumonia from the vomiting. I have had dogs successfully vomit up things they have swallowed, but given the size of the rawhide that you are describing this option makes me nervous. I would only do it at a veterinary office where they could deal with any problems immediately.
The second option is that you retrieve the rawhide from he stomach with sure or an endoscope. This is probably the safest option. The stomach is the safest part of the intestine to do surgery on. The down side is the expense. Surgery to remove the rawhide might run you $1500 or more. There would be a short hospital stay and recovery period, but this option offers the fastest safe resolution to the problem.
The third option is to feed him high fiber diet and hope that the rawhide was pliable enough that it can bend and twist it's way through the intestine and pass. I have seen this happen too.... it just depends on how thing move in there, how soft the rawhide is and how much it was chewed. If it passes you are all good, but if it gets stuck then you have to go to surgery to retrieve it. And the surgical cost and complication of removing it from the intestine once it is stuck will be higher than removing it from the stomach now. If you choose this route, watch him closely. The first sign of belly pain or vomiting and you need to head to surgery.
I hope this helps, but if you have additional questions please let me know. If not, please rate my answer.