It can be a bit risky, but have you tried doing early invades?
A lot of champions are starting Gromp or Krugs and they will smite when they start the camp to get the buff. If you steal that first camp, you will immediately get level 2 with near full health, so you can proceed to kill the enemy jungler or escape. Losing the first camp can be devastating as the enemy jungler will be missing smite and he will need to clear at least two camps (while he's still level 1) to reach level 2. He will clear slowly, be really low health, and be vulnerable to counter-jungling for much of the game.
In addition to setting the enemy jungler behind, it should leave you near full health when you begin clearing your jungle so you will end either higher health.