Run the skimmer as wet as possible. Keeps it cleaner and running more efficient.
A sensible target for "how wet" is
exactly wet enough so that the skimmer cup is full JUST when you're ready to empty it. I can get to mine about once a week, so I don't have it running full tilt....just enough so it's about full in a week.
If you're trying to get a grip on something that you think is not currently being skimmed out 100%, then skim
literally as wet as possible (don't overflow, obviously) and empty the cup 2-3 times a day if you have to.
Keep a close eye on your PO4 while you're messing with the system – raising nitrates can cause a growth spurt to happen in the tank that could use up 100% of the available PO4. Zero PO4 will bring on things much worse than a little hair algae.
In fact, consider hair algae your friend and don't work at all to eradicate it.
Consider it a polite warning from your tank that it's a bit overloaded with livestock – that's it. Not a huge deal. No battle called for.

So don't make it worse and all should be well – don't add more fish, don't start feeding 2x more, etc. A stable bio-load (or even a shrinking one) will be your friend.

(More coral
can be a stablizing factor, tho.)
The nitrates running so close to zero can be on OK thing if the lighting situation isn't too demanding and assuming flow for the corals is
nice.
Zero is not good though...corals will ultimately suffer at the hands of their competitors if they are perpetually N-starved.
If they are well fed in some type of particulate feed, this is another perfectly legitimate avenue of N-acquisition for them. This isn't usually true in a home tank, and that style of feeding is too easy to get wrong – you end up bombing your tank with waste food as much or more than you're feeding the corals you have in mind.
This is why feeding your fish (which generates particulate food AND ammonium) is really the best best option if you can do it smartly. An Eheim auto-feeder can go a long way toward helping – they are both programmable and VERY reliable! But make sure you're also feeding live foods, high quality frozen and refrigerated foods, etc in addition to any flake or pellet foods being used.
Remember above all else: When you make changes, make them small and take lots of time to observe for changes.....many changes to nutrient cycles (e.g. increases in feeding or water changes, etc) can sometimes take weeks or more to show their full effects.