Not sure if this is what you are looking for, but I will base my answer off of 2 things.
- How much faster is jumping while sprinting?
So if you want efficient, there are 2 answers both can be found in the 2 links above.
for #1, Most efficient speed, sprint-jumping travels faster even taking time to eat, so this could be considered more speed efficient.
for #2, Most efficient hunger, by itself walking (or sneaking after 1.9) would be the most efficient, besides choosing something that does not take up food (i.e. minecart tracks everywhere).
If this doesn't answer your question, then you have to decide whether you want to sacrifice speed for hunger or the other way around. If you want to make your hunger last longer, then walk everywhere. That's the simple answer.
Walking to me is the best way to go, with an occasional sprint if i get impatient. I would suggest just making shortcuts to reduce time, such as messing with minecarts and pits with water at the bottom to speed up the time it takes to travel up/down. Stairs would also make it more efficient to walk up, since you won't have to jump.
So if you want to use less hunger, then either decide to invest some time to make something like a minecart system to take you up and down the shaft, then walk/sprint the rest of the way. You can also do the same above ground, including the use of a waterway + boat to save hunger.
You can also mess with redstone, which would be more time consuming than a minecart system and still save time/hunger later. Also you could get a saddle and ride an animal down there (if you can even fit), but just another option that saves hunger. Obviously a horse or pig may not work well, but it does save hunger.