"It did not strike me that there was any recognition during the hitchhike," late-night host Seth Meyers jokes to PEOPLE in this week's issue

Credit: Seth Meyers/Instagram

Seth Meyers may be gearing up to become a father of two, but he’s admittedly experienced a few mild snafus along his parenting way.

The Late Night with Seth Meyers star talked to PEOPLE for this week’s issue ahead of his Sunday Golden Globes hosting gig, recalling a time when he and wife Alexi lost their way while exploring the great outdoors with their son Ashe Olson, 21 months.

“My wife and I and a baby strapped to my chest went on a hike and got lost pretty badly and had to hitchhike,” says Meyers, 44. “We were in Connecticut and came down on the wrong side of the mountain — hill, it was more of a hill than a mountain.”

“Everything was fine, but we were fully lost. I kept it to myself,” adds the star. “It did not strike me that there was any recognition during the hitchhike.”

Want all the latest pregnancy and birth announcements, plus celebrity mom blogs? Click here to get those and more in the PEOPLE Babies newsletter.

FROM PEOPLETV: Former Bachelorette Ali Fedotowsky-Manno Opens Up About Her Biggest Parenting Mishap

Little Ashe’s safety is Meyers’ No. 1 priority, though — in fact, worrying about his son plagues the star even when he’s asleep.

“I have a lot of nightmares about our baby being in the bed with us, even though our baby does not sleep in the bed with us,” says the Saturday Night Live alum, who announced his second child on the way on the Thanksgiving episode of his show.

“I get worried and I wake up and start looking for him and then I just grab my wife and wake her up. She does not find that endearing,” Meyers jokes. “I really never had dreams, but then you have a kid and you just have dreams all the time.”

Meyers confirms to PEOPLE that his wife will be along for the ride in Los Angeles as he hosts Sunday’s Golden Globes, and admits the work-life balance as a parent is a challenge.

“I’m in a situation where I’m lucky enough to see my son for basically an hour and a half in the morning and he’s asleep by the time I get home,” he explains. “I just try very hard to make sure the weekend isn’t spent obsessing about what’s coming up on Monday and just being in the moment.”

“I host a talk show, but my wife hosts, executive produces and writes our son’s life and is fantastic at it,” Meyers praises Alexi. “It’s just about priorities, and when you walk out of this building trying as hard as you can to leave the things you think about in this building here … They’ll wait for you.”

Golden Globe Awards - Season 75
Credit: Lloyd Bishop/NBC

Seth Meyers

RELATED VIDEO: Seth Meyers and Wife Alexi Ashe Meyers Welcome a Son

The soon-to-be father of two tells PEOPLE his second son on the way is due in the spring — and that he will be overjoyed if the new arrival is even “in the ballpark” of how great Ashe has been.

“We really like being parents and we never thought we’d stop at one,” Meyers divulges. “I think now it’ll be a conversation as to what we do next. But this, we always wanted, and we were just happy that it’s coming.”

The 2018 Golden Globes are set to air Sunday, Jan. 7, on NBC at 8 p.m. ET/5 p.m. PST.

For more about Seth Meyers’ Golden Globes hosting gig and expanding family, pick up the latest issue of PEOPLE, on newsstands now.