The closing your hand over your mouth and breathing out thing sort of works, but I think that there should be a better way.
Lick your hand, then smell your saliva.
That’s exactly what I was going to say as well.
And if it is too bad, have somebody else lick your hand and smell their saliva. 
If you don’t like the licking your own hand idea I saw someone recommend the similar activity of rubbing a metal spoon around the back of your mouth and then smelling it.
Oooh, that’s gross. How about if I just spit into my nose?
Dude, I have to break something to you - your whole mouth is full of saliva! Just covered in it! And your throat! Isn’t that just totally disgusting!?!?!
Seriously, just wash your hand afterwards, but if you find your own saliva totally gross, do you manage to kiss other people? Ever held a baby? They dribble saliva everywhere, you know, and that’s the clear, non-smelly leaking fluid. (The rest you probably don’t want to hear about.)
Am I the only one…?
Puff out your cheeks full of air, then push the air out of your mouth while simultaneously inhaling through your nose. If you aim the airstream up toward your nostrils, you can breathe your own breath in. Not as hard as it sounds.
Wake up in the morning? I can always smell it then.
It’s not the saliva that’s gross, it’s the licking your hand part. Hands are dirty and should not normally be licked.
Chalk another one up for the back of your hand- learned that in high school.
*You can wash your hands first if that’s an issue (the cleanliness of your hands).
Well, if licking one’s hand is too gross, I suppose my suggestion to hock up a loogie is out of bounds too. This is embarassing to admit*, but I’ve always found my saliva and my phlegm to smell differently – the latter smelling worse, of course.
(*not that they’re different, but that I would know such a thing)
But then, I’ve always been plagued by a chronic cough and post-nasal drip, and infected sinuses don’t tend to contribute to a fragrance of sunshine and rose petals; IOW, YMMV.
Brush your teeth with salt.