They’ve tried to grow through strategic partnerships with MGM and SLH hotels, and through acquisitions such as Two Roads Hospitality which has given them Alila, Destination Hotels, Joie de Vivre and Thompson Hotels (as well as the nascent tommie brand).
Hyatt has actually tried and failed at other acquisitions. They were in the running for Starwood, probably even offering more money than Marriott, but their complicated stock structure (the Pritzker family retains outsized control) got in the way. Hyatt tried at the last minute to acquire NH Hotels but that door was already shut. They were also a bidder for Kimpton, but IHG beat them out by $100 million. (Hyatt was right not to overpay the way IHG did.)