while I also have not found a solution to double lock on MEW, but at least you can do early prevention of phishing action in the following ways:
1. Use Chrome Browser Extension EAL or MetaMask to block malicious sites.
2. Always check the URL in Browser Bar:
https://www.myetherwallet.com. and always make sure the bar is green and it says MYETHERWALLET LLC [US], this paper does not appear and is just green like this picture
3. Do not easily believe in messages sent via email, reddit, twitter, slack and other media.
4. Always highlight on the link and do not click on it, when we highlight the link will appear the address at the bottom left corner of the browser, we make sure whether the address is trusted or not.
5. Never once provide confidential information on websites that you have accidentally clicked it.
6. Do not click any ads on the website, and if necessary install adblocker to avoid other things.
Actually, I have never been confronted with phishing and even I just found out. I can not advise and input, but I recognize that this topic is really useful to me. And I can do the steps you give in this topic. Or maybe there is another solution to secure tokens / coins in MEW?