If you want to use a mixing service, you will need a reputed one that you can trust, if you do not trust any, then do not use any mixer. But after having less privacy like your bitcoin addresses are connected together (maybe) transaction, this would be available for the public as the blockchain is transparent. This is one of the reasons someoen can decide to use a mixer, even if the addresses are known to the mixer, the public will not know the owner of the old address is the owner of the new address which is the function of a mixer.
If you use tor or use a public WiFi service then you'll be much more hidden.
Public WiFi is not secure, it would be better with Tor for anonymity.