Rather than smoke, I suspect that you are seeing steam, and the sweet smell would seem to confirm that.
Because coolant has a sweet-ish smell (and a sweet taste, which is why this toxic substance has to be kept away from animals), I am almost certain that you have a small amount of seepage from the cooling system, and that what you are seeing and smelling is steam from that system. The easiest way to confirm my theory is by checking the level of the coolant in your radiator. (Just be sure to open the radiator cap when the engine is COLD!) If the radiator is not full, that would likely confirm that there is a leak from somewhere in the cooling system.
If the “smoke” is coming from the bottom of the engine, then one possibility is that one of the freeze plugs in the engine block has been perforated by rust. If you don’t take care of this situation very quickly, you will likely run the cooling system dry, and you could wind up frying your engine very rapidly when it overheats.
There are other possibilities, of course, but I really think that you need a new mechanic. This is not something that should be very complex to diagnose, and that statement about needing more time to figure out the problem points toward a mechanic who is either not very skilled or is not very interested in helping you.
Edited to add:
If it turns out that you have a coolant leak from a rusted-out freeze plug, that is an indication that the coolant has not been changed every few years, as it should be, and that the cooling system is likely filled with rust and other residue. In a situation like that, in addition to replacing the freeze plug, you will need to have the cooling system flushed before adding fresh coolant in the correct ratio. And then, you will need to remember to change the coolant every 3 years.