While, generally speaking, you are limited to ten floors of a dungeon per day, each dungeon is split into 3 separate Tiers. Each of these tiers starts at a different depth - 0, 8, and 18, So if you have Tier 2 Clearance, you can start at floor 8, and go all the way to floor 20 (no energy costs to access floors 13 and 18).
Likewise, if you start at floor 18 (tier 3 clearance), you can make it all the way to the core, Level 29, and still have 10 energy left over.
The one downside of this is that starting at Tier 2 or 3 costs Crowns (the in-game currency) -- 200 or 500 respectively. However, if you work your way through one gate sequentially - getting to floor 8 the first day, getting to floor 18 on the next, etc. you will avoid having to pay the cost in crowns.
Since gates have an 8-day cycle, this means you will be able to delve all the way to the Core 6 out of 8 days, avoiding the 'tax' for jumping directly into the deeper tiers. Alternatively, just jump directly into Tier 2 and Tier 3 without clearance; the Crown tax isn't that bad, and you will generally recoup your entry fee within the first floor or two.