Chase has some of the best cards out there, but I think you're wise to space out your apps.

Thanks for so clearly stating your goals and nearby airports, that makes it so much easier to offer advice.

I'd give my vote to SPG AX and Barclay's Arrival+ based on your goals.

SPG has Nights and Flights packages that you should look into. The nice thing about that is for 60,000 pts (Cat. 3) or 70,000 pts. (Cat.4), 50,000 of the redeemed points go to any of Starwood's airline 1:1 partners (e.g. BA, AA, many others) and you get 5 nights at a Cat. 3 or 4 Starwood property. It has a $65 annual fee, but with AMEX Offers I recoup that fee easily. The Cat. 3 and 4 properties may not be as high end as Hyatt but you'd have another Chase card if you apply for the Hyatt CC.

Barclay's Arrival+ MC is a great card, especially if you're looking at foreign travel since it is a chip + PIN card (not chip + signature) which is widely accepted worldwide and there's no foreign transaction fee. The annual fee at $89 is a little steep, though the sign up bonus covers that for a few years.

I too am looking at which card I should apply for next. Decisions, decisions.