You can use Milorganite in the summer if you want, but it isn't necessary. It is relatively low in N, so you won't get to much forced top growth like you do with synthetics, so it's pretty benign. You will get some iron from it, which will help green up the grass, as well as some increase in soil microbes. Your grass will do best getting one inch of water a week, applied in one watering. Of course you can't control the rain, but it is best to water between 12-6 AM.

For watering, watch the grass, not the calendar. When the grass needs water, water it. So much depends on rain, humidity, cloud cover, dry winds, morning dew, shade, soil type, and grass type and height. For most people, though, summer heat means weekly watering. The arrival of your summer heat season is not marked on the calendar.