On density of old sets in Prikry type extensions

Every set of ordinals of cardinality κ in a Prikry extension with a measure over κ contains an old set of arbitrarily large cardinality below κ, and, actually, it can be split into countably many old sets. What about sets with larger cardinalities? Clearly, any set of ordinals in a forcing extension of a regular cardinality above the cardinality of the forcing used, contains an old set of the same cardinality. Still cardinals in the interval (κ, 2^κ] remain. Here we would like to address this type of question.