How to run a dangerous 13th Age game?

There is no doubting it, 13th Age characters are powerful/heroic. Is there anyway I can make the game seem a little more dangerous and gritty? I know it will mostly rely on playstyle but I want some of the mechanics to reflect this. What are your suggestions? I was thinking just a possible tweak on damage output from monsters and NPCs. Also possibly limiting the ability to rally.

I normally run encounters 1 level above the baseline per tier from the core rules. This also happens to increase damage output from monsters, as well as their durability.
I like Rally as it is, as it pretty much allows to play without a dedicated healer necessarily.
What made a difference for me is when I started using the 'full heal-up' rules as they are written, so completely independent from the daily clock: it's every 4 battles (or when the GM says so) that the party gets a full heal-up, no matter how much they sleep in between. This doesn't make encounters more challenging per se, but it puts pressure on Recoveries management and adds tension to fights.

I thought the "full heal-up" rules were a bit wonky to begin with. But I've heard mostly good things from them. It's just hard to escape the full-rest style of things 4e implanted in my brain.