Dry socket healing bad breath

[Emily]

I developed a dry socket after my wisdom tooth extraction. Had it packed about four days ago. The dentist said let the packing fall out as it heals. About half of the packing has fallen out so far but now my breath is revolting. Hardly any pain other than a minor nerve pain. I keep my teeth clean. Brush 3-4 times a day, floss, rinse with salt water AND alcohol free mouthwash but the smell and taste is getting worse no signs of infection either. Any tips on how to get rid of it?
Edit: apparently the breath doesn't smell but the taste is hideous