This happened about three years ago in South London, with everyone being surprised at the volume of 13+ applicants and pupils from several preps in the area being left without places. One cause may have been an ongoing shift away from boarding.
I think one of the existing less selective schools, spotting potential fee paying customers decided to add a bulge class. It will also have helped that Radnor House, which had been set up with hedge fund money, was opening its doors with a 13+ entry point. And since then a further new secondary has opened.
Your Head should phone other preps to find out if it is a general problem. If it is they can use their contacts to persuade senior schools that the first to offer a bulge class should pick up most of the business.
If it is just your school then try to find someone experienced and sensible. At 11+ the new Head at DDs school was pretty useless, but the Deputy was happy to help. (Looking back I think she was polishing her CV - she has done deservedly well since.) Then as other have suggested, read the Good Schools Guide, pick up the phone, perhaps employ an educational consultant though you would be paying for something that the Prep should be doing).
I would be tempted to write to the Governors, politely, suggesting they might offer you a discount on next term's fees to cover your additional costs. This might spark them into putting pressure on the Head. And they might not be aware of the problem yet.
Good luck. It must be awful for your son.