For weight gain, free choice/constant access hay is the primary starting point. I understand you are boarding, but this is a need that you will have to find a way to accommodate. It may mean paying more or going out yourself, etc, but your horse's health is your primary concern at this point. There is no such thing as a "standard" flake of hay when it comes to weight. One person's idea of a flake can be very different from the next - that is why weight hay, when it isn't offered free choice, is important.
From there, as pointed out, weighing all feed is a must. You can't know where to add/subtract feed if you don't know what you are feeding to begin with. A "scoop" could be a coffee can, a 2 quart scoop, a 3 quart scoop or a measuring cup -- at the very least, you need to know what sized scoop is being used and, from there you can estimate the weight of your feeds using the standard averages for various feeds that can be found online.
Boarding is really no reason you can't know these things - ask what size the scoop is that is being used and/or take one feedings worth of each feed and weigh it (either at the grocery store, a home scale, the barn's scale, etc).