Part 1 - Drones
Part 2 - Tanking
Part 3 - High Slots
Beyond the usual cargo hold and drone bay, capitals have ship maintenance arrays and corporate hangers. On a supercarrier, the ship maintenance array is 2.5 million meters cubed (enough to hold 5 battleships) and the corporate hanger is 50,000 m3, bigger than an industrial ship's hold. Beyond logisitical operations, these facilities must be put to good use.
With the additions of fuel bays, these spaces can be dedicated more to utility items needed in the course of a battle. Nanite Repair Paste for fixing overheated modules is a must for one thing. Another is having some boosters on hand. For a passive buffer shield tanked Wyvern none of the boosters is particularly inviting except perhaps the Mindflood booster which boosts capacitor energy, useful for powering shield and energy transfers. If you throw on a shield booster, the Blue Pill booster which boosts the amount of points you get from a booster is best. For armour tankers, Exile boosters improved armour repairers.
Beyond having a back stockpile of jump fuel, the best thing to have in the hanger for other pilots to access is ammo, charges, and strontium for carriers' triage mode and dreadnoughts' siege mode. Also, you might keep a set of extra modules for your own ship for hot-swapping during an operation. For example, you might be configured for POS repair and you get hot-dropped by an ambitious enemy fleet. You just access a wingman's carrier's ship maintenance array to allow you to swap out modules to a more aggressive/defensive fitting.
Ship Maintenance Array:
Holding five extra battleships is not going to help much because your battleship pilots are most likely in a snipe fleet and probably not as close to the action as a supercarrier is. But what will help? Extra Tech I tackle frigs and cruisers are a good idea as the support fleet usually gets chewed up the most; some logistic ships are popular, both Tech I and Tech II; covert ops and recons; a cyno ship or three for calling in reinforcements; Interdictors as they suffered as much as interceptors do to enemy fire; basically anything that can be spared and is useful to the fleet in the heat of the battle when a station might be too far away. You can fit a lot of smaller ships in a supercarrier, so make use of that space.