Like Gmail's update, that's out of your control.
But what IS in your control 100% is WHAT you send to your email list.
For me, here's what has worked amazingly well despite Google's new promo tabs.
In every one of my emails, I try to offer such valuable, life changing information... that people on my list will seek me out.
They'll click through 100 tabs to find my stuff, because it helps them.
Not just once in a while, but every single email I send, I want it to be so good that people think "Wow, I should have just paid for that"
What I do with each email is, I focus on one specific problem my market has, and I give away a valuable tip/technique/strategy to solve one of their problems.
Every single email.
Over time, if every one of your emails is so helpful and informative, your prospects will search for it and find it.
But... be just another marketer, and they won't bother.
You want your list to welcome you, to WANT to hear from you.
And the best way to do that is every single time you get ready to click "send", ask yourself... "does this email contain an exact tip or piece of advice that my list can go out and use, and get results."
If you can say "yes", click send.
that's how I've managed to deal with the Gmail update... I try to write such incredibly valuable emails, my list will seek me out and try to find me, no matter what tab i fall under.