There appears to be every reason for Josh Hill to exhale and relax a bit, to survey his good fortune, to relish the production he has had and bask in the knowledge that bigger and better things are on the horizon.

After all, perhaps no New Orleans Saints tight end stands to benefit more from the trade of Jimmy Graham than does Hill. Likely, no one should have experienced a bigger jolt of confidence from the fact that New Orleans didn't pick a tight end in the draft, and didn't seek a higher-profile veteran partially based on the belief that Hill couldn't handle an expanded role.

No tight end on the roster appears to be better aligned for a statistical spike and increase in stature than Hill, the once-undrafted rookie from Idaho State who will enter his third NFL season tied with Marques Colston as the Saints' leading returning player in touchdown receptions, after he caught five last year.

Thus, there's the likelihood that significantly more eyes will be trained on Hill this week during the Saints' three-day minicamp, which will be held Tuesday through Thursday at the team's practice facility in Metairie.

All three workouts will be open to the public, each day from 11 a.m. through 1:20 p.m.

Hill only needed 14 receptions in order to post his five touchdowns. But while his touchdowns-to-receptions ratio is sure to decrease this year, his opportunity to be on the field in myriad offensive sets figures to increase exponentially.

"It just depends on how much two tight end sets we're in, how much sub (packages)," Coach Sean Payton said. "I couldn't say specifically that he is going to have 25 percent more playing time but certainly his playing time will increase.

"We utilize multiple tight end sets and obviously that'll affect him with his number count (of snaps). He has good position versatility. We think he is a guy that can run and stretch the defense. A lot of it will be by game plan and what we are trying to do."

The high regard that Payton and his staff have for Hill has been evident from the beginning. The go-by-what-we-see staff did exactly that when Hill earned a roster spot two years ago, behind Graham and Benjamin Watson.

Hill didn't receive an invite to the NFL Combine, but Payton insists that if Hill had, his times and ability would've all but assured that he wouldn't have gone undrafted. And Hill hasn't given anyone a reason to not believe that deduction.

Fourteen catches for 176 yards and five scores last year, and career totals of 20 catches for 220 yards and six touchdowns, only hint at the potential of the 6-foot-5, 250-pounder who is gaining confidence and comfort in his role.

"In the OTAs, I've gotten a few more reps with the first-team offense than I have in the past," he said. "My mind-set really is the same that it has been, being aware of the opportunity but just coming in and doing what they ask me to do.

"I don't know their reasoning for (not drafting a tight end), if it's their confidence in the tight ends in the room or not. I'm just excited I'm getting the opportunity and the reps and I'm just looking forward to growing my role, and understanding my role on the team."

Apparently, he's ready for the workload. His skills are impressive.

"He's doing good," safety Kenny Vaccaro said. "Me and Josh came in together (Vaccaro was the Saints' first-round pick in 2013). Josh is a hard worker, can play every special team – I've never seen a tight end that can contribute as much as he can.

"He's really athletic. He's getting better and better. We go at it every day, just like me and Jimmy did. We've gotten to know each other real well."

Said Hill: "Kenny is a very competitive guy. We line up across from each other and we know we're going to give each other the best shot we have. There's always that competition out there.

"It makes you a lot better because you're playing against some of the best guys in the NFL every single day, so you don't just see it on Sunday, you see it at practice.

"I guess it's kind of hard to envision, (considering) where I came from, being undrafted. You always have goals and mine were always just to increase my role, just increase what I can do because the more you can do, the more valuable you are. I really try to do whatever they ask me to do and it's growing, and I'm excited about it.

And when Hill references being able to do more, it's not just talk. He earned a roster spot as a special teamer; he can contribute to every special team unit, and expects to continue to do so.

"I'm still, every day in practice, getting those repetitions with special teams as well," he said. "If they want me to play special teams, I'm going to play special teams. It's really just whatever they ask me to do.

"I enjoy playing special teams. I'd never really played (special teams) until I got here, but it's a strong culture in the special teams unit and I really enjoy that part of it."

That's the job he knows best, but he's learning the other one on an accelerated curve. His workload figures to increase, as will his production.

And there's every reason for him to believe it will.

"I understand the offense a lot more," Hill said. "Being around the team for three years now allows you to play faster, be on the page with other players. Getting that play time and more exposure last year definitely helped."