Some problems occur when your Internet security program was set to trust the previous version of Firefox, but no longer recognizes your updated version as trusted. Now how to fix the problem: To allow Firefox to connect to the Internet again;

• Make sure your Internet security software is up-to-date (i.e. you are running the latest version).
• Remove Firefox from your program's list of trusted or recognized programs. For detailed instructions, see

Configure firewalls so that Firefox can access the Internet. {web link}

Thanks, but Firewall has been turned off...

Type about:preferences#advanced<Enter> in the address bar.

Under Advanced, Select Network.

[56+] Type about:preferences#general<enter> in the address bar. Look for Network. +++++++++++++++++++++ Look for Configure How Firefox Connects and press the Settings button. If you are using a proxy, make sure those settings are correct. If there is no proxy, first try No Proxy. If there is a problem, then try System Proxy.

Thanks. I have tried both no proxy and system proxy, but didn't work.

I have discarded /Users/xxxxxx/Library/Application Support/Firefox including Profiles and relaunched Firefox, but still it doesn't connect anywhere. No Addons.

Start your Computer in safe mode with network support. Then start Firefox. Try Secure websites. Is the problem still there?

http://encyclopedia2.thefreedictionary.com/Linux+Safe+Mode Starting Any Computer In Safe Mode; Free Online Encyclopedia

In Safe mode, Firefox behaved in a strange way in that the GUI is a bit broken, no toolbar, web page zoomed like 200% etc. But it connected to the Internet anyway.

Then I restarted Mac in the normal mode. I found I'm no longer using LongMeIn Hamatchi, so I uninstalled it. Then, the Firefox just worked as expected.

It doesn't seem to be totally making sense, but I think that Safe mode helped somehow.

Thank you so much for your support!!!