The HMC406 is intended to be used from 5-6 Ghz. The S11 seems to be optimized for 4.4-5.1 GHz based on the S-parameters provided. Can the S22 be matched to 4.4-5.1 GHz?
Parts like the HMC406, HMC407, etc. were intended for specific frequency ranges and have had their application circuits optimized for those ranges. You may be able to alter the output tuning components to obtain some improvement in the gain, return loss, etc...possibly at the expense of other parameters such as the linearity. Perhaps start by increasing the value of shunt capacitor C5 to shift the response lower.