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There is no way that just moving everything from your old HDD to a new HDD would cause the problems you're having (unless there's some software that was dormant on your old HDD that got triggered by whatever event, to push ads, but that seems unlikely).

You're going to have to tell us: 1) Where did you buy your HDD from - was it new? 2) What steps did you take to move programs/data from your old HDD to the new HDD? 3) What software have you installed and or suspicious downloads have you done since your new HDD was in place? 4) Have you loaned the computer or left it unattended since you installed your new HDD?

(NOTE: Some sites will pretend to be legitimate but will not be - adware sites saying you need to update your Flash plugin are one example.)

You can try downloading and running EtreCheck and posting the results here.
https://etrecheck.com/

 

So, given that scenario, where do you have your firewall? If your home router/whatever doesn't have a firewall or if you use the computer on wifi networks where the firewall status is unknown, then you should have the internal firewall turned on. That's in System Preferences -> Security and Privacy Pane (first row) -> Firewall. Maybe that setting wasn't turned on when you moved over to the new HDD? I don't know if that's a typical way to get adware but is something to check anyway.

 

Generally not. A couple of articles to read:

https://support.apple.com/en-us/HT201642

http://www.howtogeek.com/205108/your-mac’s-firewall-is-off-by-default-do-you-need-to-enable-it/

 

As I mentioned, I don't think the type of attack that a firewall prevents is the way most adware is installed. But the firewall setting is probably the most obvious setting where the less secure setting is the default and which may have been turned off if you didn't do a full restore. Glad you got rid of the adware.

 

You need this FREE app! Download it and run it. ALL ads gone!!

https://www.malwarebytes.org/antimalware/mac/