Avoiding this scenario is one of the primary reasons you should obtain multiple written estimates before signing a repair contract. Having more than one contractor examine the damage reduces the likelihood of having hidden damage and gives you a robust view of what the repairs are expected to cost. If you did have written estimates and the chosen contractor still found hidden damage that escaped notice through the estimate process, your options are going to be limited.
Contact your insurance company and explain the situation. The homeowners insurance company may send out an adjuster to assess the damage and may renegotiate the amount of the settlement. It is important that you contact the insurance company as soon as the additional damage is found, because you may not be eligible for renegotiation after the insurance company has paid the initial settlement. In that situation, you would be responsible for paying the additional costs out of pocket.
Depending on the cause of the hidden damage, you may be able to file a second claim. This method will only work if the newly found damage is not a result of the peril which caused the initial damage. In this situation, you will need to obtain new estimates and have the damage assessed by an insurance adjuster to verify that you have a legitimate claim. Keep in mind that any deductibles or other costs specified in the policy will be your obligation before the new claim can be settled, just as they applied to your initial claim. The danger here is that filing too many insurance claims in a short period of time can have unwanted consequences, including higher premiums or even having your policy declined when renewal time rolls again.
It may also be a good idea to have the hidden damage verified by another contractor. There are cases where a contractor "invented" hidden damage or caused it themselves in order to beef up the price of repairs. This is not considered ethical, but as the ancient saying goes, "Let the buyer beware."