You can consider applying for a credit card which is not an airline credit card. For example, it can be the Discover it® Miles credit card. The card will allow you to earn 1.5 miles on all your purchases. All earned miles can be redeemed for airline tickets with any airline, hotel rooms, rental cars or cash. The miles you will earn are basically cash back rewards, so you can redeem your miles for cash and cover the expenses that are not covered by your Visa Platinum rewards. Discover cards do not have annual fees.
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All rates and fees, and other terms and conditions of the products mentioned in this article/post are actual as of the last update date but are subject to change. See the current products' Terms & Conditions on the issuing banks' websites.