For a double dip, it's usually recommended to start applying early during regular business hours of a non-holiday weekday in case you need to call Chase reconsideration ("recon") or verification departments. Chase Credit Card Reconsideration. Personal: 1-888-270-2127, open: 8am-10pm ET Monday to Friday. 8am-8pm ET Saturday & Sunday. The verification department may have different hours.
For personal cards, be sure you're not running afoul of the 2/30 rule: you can only have two Chase credit card applications every 30 days or you'll likely be denied.
Apply for the card you want
first. If you're not instantly approved and perhaps got the we'll let you know in 30 days message, call the automated status line at
1-800-432-3117 to check the status of your application. If you still aren't approved, call recon at
1-888-270-2127 to get it approved. Mention you’re calling (only) about the first application, and ask what can be done to move the application towards approval. They'll likely ask you identity verification questions.
After that first card is approved, use a different internet browser or an incognito window for the second application (and you can even clear the history), so Chase doesn’t get your cookies crossed and think you’re submitting a duplicate application in error. If you're not insta-approved, you can try to wait it out by checking the automated status line after an hour or so, but if you're not approved then call recon again to get the second application approved on the same day as the first approval. If you have to call recon, mention that you are calling about the
second card only, and ask how you can help get it approved. You don't want recon realizing you've just been approved for an earlier Sapphire. It's likely an issue with needing to move around credit lines in order to have enough available credit to open up a new card. (Not sure if you have any Chase cards from which you can move credit to a new card. You won't be able to move credit from the newly approved Sapphire card to a second card, nor should you try.) If the rep does mention the other Sapphire card, and asks why we need the two cards, a lot of us just say we want to separate expenses.
Good luck!