The only time Kaden Smith has played at Levi’s Stadium, he put on quite a show.

His Stanford team lost to Sam Darnold and USC 31-28 in the Pac-12 title game in 2017. But Smith caught an 11-yard touchdown pass from K.J. Costello late in the third quarter and then — with a couple of minutes left in the game — he made a spectacular leaping catch between two defenders for a 28-yard touchdown.

The 49ers, no doubt, paid attention to those plays, especially because they came in their own home. When they drafted the 6-foot-5, 252-pound tight end in the sixth round last week, however, he was surprised.

Before the draft, he said Friday before a rookies minicamp at Levi’s, “I think I only talked to the 49ers once.” After the pick, he was “super excited. I’m 20 minutes down the road from Stanford.”

Along with four other draftees, he signed a four-year deal, the club announced.

The only 49ers game he has seen in person was the 2017 opener when defensive lineman Solomon Thomas and another former Stanford standout, Carolina Panthers tailback Christian McCaffrey, made their NFL debuts.

Also making his debut in that game was 49ers tight end George Kittle, who last season broke Travis Kelce’s NFL’s single-season record for receiving yards by a tight end with 1,377 and made the Pro Bowl.

“I saw what he did last year,” Smith said, “and I’m excited to come and learn under him and eventually fight for a spot. I’m not really sure what my role is yet, but I’m just going to play football and enjoy it and have fun and keep learning.”

He’ll also compete with 30-year-old Garrett Celek and Ross Dwelley, who entered the league last year as an undrafted free agent from San Diego.

“Kaden is the most complete tight end coming out in this year’s class,” said Morgan Turner, who’s somewhat biased because he’s Stanford’s tight ends coach. “His ability to make tough catches is rivaled by very few. He also wants to be part of the run game, which very few guys want to do these days. You can put him at the point of attack and trust him in the run game. Kaden really embraced that and did a great job with it.”

Nevertheless, Smith was only the 13th tight end picked, possibly because his 40-yard dash time (4.9 seconds) at the NFL combine is considered slow for players at his position. He didn’t run the 40 at Stanford’s pro day.

“I figured a team would draft me based off my skills,” he said. Pointing to his 47 catches for 635 yards last year, he said, “I get open. … Separation isn’t really a problem for me. And blocking, too — understanding (about) staying low, going against some bigger guys.”

Stanford fans are excited to see what Smith is going to do in the NFL. So are other former Cardinal tight ends like pros Zach Ertz, Austin Hooper and Dalton Schultz, all of whom texted congratulations to him on draft day.

“One of the reasons I went to Stanford was they had an NFL offense, with NFL lingo,” Smith said.

The 49ers’ playbook is “very similar to what we had at Stanford — same formations and similar play calls and concepts,” he said.

All in all, he’s delighted the 49ers took him. “I was very thankful,” he said. “I love this area. It’s great to be back. All my friends are out here.”

Tom FitzGerald is a San Francisco Chronicle staff writer. Email: tfitzgerald@sfchronicle.com Twitter: @tomgfitzgerald