July is perhaps the best month. The problem with the Moy is that around 60% of it is not good fly water, very slow flow. That being said, there are some nice beats, and the run of grilse can be a very good one. If going there, please try the ridge pool and the cathedral beat, at Ballina, on the estuary. This is tidal water, you loose half of the day because of high tide, but at lower tide it is good fly water. If the season has been dry, you can fish over countless numbers of fish, which you can see from a bridge.
July is the best month for the Galway Weir beat, in the middle of the town. If just one or two gates are open, very good fly water. You have to book in advance for one or two days, and take your chance. More than 3 gates open and it is not possible to swing a fly there to catch a salmon.
July and the first half of August is big time in the Spey. Very good fly fishing river. There is good association water at Grantown, perhaps the best in all of the UK and Ireland.