Practical deployments of key encapsulation mechanisms (KEMs) may entail large servers each using their public keys to communicate with potentially millions of clients simultaneously. While the standard IND-CCA security definition for KEMs considers only a single challenge public key and single challenge ciphertext, it can be relevant to consider multi-target scenarios where the adversary aims to break one of many challenge ciphertexts, for one of many challenge public keys. Many post-quantum KEMs have been built by applying the Fujisaki-Okamoto (FO) transform to a public key encryption (PKE) scheme. Although the FO transform incurs only a few bits of security loss for the standard, single-challenge IND-CCA property, this does not hold in the multi-target setting. Attacks have been identified against standards-track FO-based KEMs with 128-bit message spaces (FrodoKEM-640 and HQC-128) which become feasible if the adversary is given many challenge ciphertexts (say, 2^64). These attacks exploit the deterministic encryption induced by the FO transform which allows the IND-CCA experiment to be reduced to a search problem on the message space, which in some cases may not be large enough to avoid collisions between pre-computation and challenge values. A cost effective way to amplify the hardness of this search problem is to add a random but public salt during encapsulation. While revised versions of FrodoKEM and HQC have used salts, there has been no proof showing that salting provides multi-ciphertext security. In this work, we formally analyze a salted variant of the Fujisaki-Okamoto transform, in the classical and quantum random oracle model (ROM); for the classical ROM, we show that multi-target IND-CCA security of the resulting KEM tightly reduces to the multi-target IND-CPA security of the underlying PKE. Our results imply that, for FrodoKEM and HQC at the 128-bit security level, replacing the FO transform with the salted variant can recover 62 bits of multi-target security, at the cost of a very small overhead increase.