Bairstow, who has played 69 Tests, has passed 50 with the bat just twice since the Sri Lanka Test in November 2018.
“He’s had a very busy summer, it’s an opportunity for him to step back, reset and come back better,” Smith added.
Uncapped Matthew Parkinson, Dom Sibley, Zak Crawley and Saqib Mahmood have all been called up.
Wicketkeeper Bairstow averages 20.25 in Tests this year, so Jos Buttler will take the gloves.
- England in New Zealand – full tour schedule
But Smith told, he does not want to restrict Bairstow to a specialist batsman role on his return.
“I wouldn’t want to reduce his options, I want to expand them,” insisted Smith.
“With Jonny, someone who averages 50 in first=class cricket, non-Test match cricket, they are exceptional numbers. But I think what we’ve seen recently in an England shirt is not a reflection of what he can do.”
The Yorkshire player has been included in the squad for the T20s in New Zealand, which is led by Eoin Morgan.
Warwickshire’s Sibley and Kent batsman Crawley will compete for places at the top of the order.
Pace bowler Mahmood and leg-spinner Parkinson provide bowling cover.
England play two Tests in New Zealand, the first of which begins on 20 November, after a five-match Twenty20 series.
They then move on to South Africa in December before two Tests in Sri Lanka in the spring.
Sibley, 24, is the only batsman to make more than 1,000 runs in Division One of the County Championship this season, while 21-year-old Crawley is highly regarded and has made 820 runs this summer.
Mahmood and Parkinson, both 22 and of Lancashire, have 21 and 19 wickets respectively in Division Two.
The pacy Mahmood gives back-up to a fast bowling unit that is without the injured trio of James Anderson, Mark Wood and Olly Stone, while Parkinson will be number two to left-arm spinner Jack Leach.
Anderson, England’s all-time leading wicket-taker, has still not recovered from the calf injury that curtailed his part in the Ashes series.
Surrey batsman Ollie Pope, who played two Tests in 2018, is recalled. With no specialist keeping cover for Buttler, Pope could take the gloves if required.
Morgan, who captained England to their World Cup triumph in July, has ended speculation over his immediate future by being named skipper.
Morgan said he would only continue as captain if his troublesome back allowed him to do so.
With England preparing for a Twenty20 World Cup in Australia next winter, a number of the squad that won the 50-over version are not included.
Jason Roy, Test captain Joe Root, all-rounders Ben Stokes and Moeen Ali, wicketkeeper Buttler, and pace bowlers Wood, Jofra Archer, Liam Plunkett and Chris Woakes all sit out.
Mahmood and Parkinson join exciting Somerset batsman Tom Banton and Worcestershire seamer Pat Brown in the squad.
Only Joe Denly, Sam Curran, Mahmood and Parkinson are included in both squads as England rotate their players at the end of a busy year.
England squad for two Tests in New Zealand: Joe Root (Yorkshire) captain, Jofra Archer (Sussex), Stuart Broad (Nottinghamshire), Rory Burns (Surrey), Jos Buttler (Lancashire), Zak Crawley (Kent), Sam Curran (Surrey), Joe Denly (Kent), Jack Leach (Somerset), Saqib Mahmood (Lancashire), Matthew Parkinson (Lancashire), Ollie Pope (Surrey), Dominic Sibley (Warwickshire), Ben Stokes (Durham), Chris Woakes (Warwickshire).
England squad for five Twenty20s in New Zealand: Eoin Morgan (Middlesex) captain, Jonny Bairstow (Yorkshire), Tom Banton (Somerset), Sam Billings (Kent), Pat Brown (Worcestershire), Sam Curran (Surrey), Tom Curran (Surrey), Joe Denly (Kent), Lewis Gregory (Somerset), Chris Jordan (Sussex), Saqib Mahmood (Lancashire), Dawid Malan (Middlesex), Matt Parkinson (Lancashire), Adil Rashid (Yorkshire), James Vince (Hampshire).