Start inside, not on a ladder
The safest first check happens indoors. Walk the top floor and look at ceilings and the tops of exterior walls for a new stain, a damp patch, or paint that has started to bubble. Check the attic if it is accessible, using a flashlight to look for wet insulation, dark staining on the underside of the deck, or daylight coming through where it should not.
None of this requires climbing onto a wet roof right after a storm, which is when most fall-related injuries during storm cleanup actually happen.
What to look for from the ground
From the yard, scan the roofline for shingles that are visibly missing, folded over, or sitting at an odd angle compared to the rest of the slope. Look at the gutters for anything unusual: sagging sections, a downspout pulled loose, or debris and granules built up that were not there before. A pile of dark granules at the base of a downspout after a hailstorm is a sign worth noting, even without visible shingle damage.
Trees and debris
Any limb resting on the roof, even a small one, is worth a call before it is moved by anyone without the right equipment, since removing it wrong can do more damage than the branch itself. Clear smaller debris like leaves and twigs from valleys and around chimneys once it is safe to be on the ground near the house, since trapped debris holds water against the shingle longer than it should sit there.
When it is worth a same-day call
An active drip inside, a large section of missing shingle, or a tree limb on the roof are all reasons to call the same day rather than wait and see. Cosmetic-looking issues like a little granule loss or a single lifted tab can usually wait a day or two for a proper look, but noting the date the storm happened is useful either way if an insurance claim ends up being part of the conversation.