Inside Taylor Swift and Joe Alwyn's Undercover Romance

Taylor Swift and Joe Alwyn continue to keep their new romance private—a contrast to two of her most recent previous celebrity relationships—while the singer also continues to avoid the spotlight, sometimes by taking extreme measures.

In recent months, the 27-year-old singer has been traveling back and forth between England, where he lives, and Nashville, where her family lives and where she has been recording her new album. In addition, the 26-year-old British film and theater actor has himself traveled to visit Swift in New York—twice—and the two have also kept their visits on the down-low, a source told E! News exclusively Thursday.

It comes months after a more public, highly scrutinized and short-lived relationship with actor Tom Hiddleston, which was exposed after the two were photographed kissing near her Rhode Island beachfront home and which came after a highly publicized one-year romance with Calvin Harris.

The two were driven in a chauffeured car with her bodyguards, the source said, adding, "They didn't want to be seen by anyone and were ducking down in the car and wearing hoodies."

It appears they will be traveling for a while, as the two had a "ton of luggage with them," the source said.