Ty Burrell Shares Family Trips That Have Gone 'Sideways' — Including a ‘Memorable’ One When His Daughter Got Injured (Exclusive)

Vacations aren't always the most relaxing for Ty Burrell.

In an exclusive interview with PEOPLE, the Modern Family alum, 57, recalls family getaways that have gone wrong in the past, including a recent one when one of his daughters got injured.

"We were biking in the Dolomites [mountain range in Italy], like really hard biking — at least for me, maybe not for an actual bicyclist — and a bee got in front of her, and she took her hand off the handlebar, went to swipe it and she flipped over," Burrell says.

Describing his daughter as "a really tough kid," the actor adds that she persevered and "got back on her bike, but has had a chipped tooth ever since."

"She's never [gotten it fixed], I think, because that trip was so memorable," continues Burrell, who has partnered with Chase and United Airlines to announce new and enhanced travel benefits for United MileagePlus card members. "I think in some ways, it's one of her proudest memories, because she got back on the bike and finished the ride."

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Burrell and his wife, Holly Burrell — who wed in 2000 — are parents to two children: daughters Frances and Greta.

The family loves to travel together, but Burrell tells PEOPLE they always remain prepared to expect the unexpected. "I don't think we've ever had a trip that didn't end up having things go wrong, sometimes really sideways," he says.

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Burrell and his family have frequented many spots across the globe, including parts of Africa, which he says they would love to "see the rest of," and South America, which the star calls "incredible."

On the family's travel bucket list, however, remain a few standout spots, including New Zealand. "As a person who fishes, that's sort of a legendary place that's high up there for that," Burrell says.

No matter where his travels take him, he always knows the importance of spending smart, which is where what he describes as an "organic" partnership with Chase and United comes in.

The two companies announced on March 24 that new and existing United MileagePlus card members would be receiving "new and enhanced benefits designed to make flying the world’s largest airline even more rewarding, valuable and personalized."

New benefits, Chase and United said in a press release, include airline and rideshare credits, car rental and hotel awards and United Club access, as well as exclusive offers from brands like Instacart.

Card members will also be able to earn toward higher status levels and receive travel discounts when miles are redeemed to fly to hundreds of destinations across the world, "including more than 70 that no other U.S. airline serves," the release added.

"My family and I, we've always enjoyed flying United, and these cards are just genuinely helpful," Burrell tells PEOPLE. "I mean, to put it just as simply as possible, they're really trying to make travel simpler and easier."