Town, who beat Reading in the Championship play-off final in May, have been tipped by many to make an immediate return to the second tier.
The Terriers start the season with a trip to Crystal Palace on Saturday.
"People didn't expect anything from us last season and look what we achieved," the defender told BBC Radio Leeds.
"We are more than happy for people to write us off and more than happy to be the underdogs, that's absolutely fine. But as long as we go about our business in the right way, which I'm sure we will, then I see no reason why we can't give it a good go.
"We've got a good squad and good togetherness, and all of the attributes that we had last season I think will continue into this season.
"We know it will be tougher but we see no reason why we can't stake a claim to definitely stay in the league and progress higher up it."
The last Championship play-off final winners to avoid being relegated were Crystal Palace in 2013-14.
Queens Park Rangers, Norwich City and Hull City have all gone straight back down after winning promotion at Wembley in the last three seasons.
Smith, 25, missed much of Town's pre-season preparations with a foot injury sustained in the win over the Royals but is hopeful he will be able to feature against Palace.
"I feel good. It took a while but the doctors and physios said it would take time but I was desperate to get fit as soon as possible," he added.
"It looks like I'll be available for selection so I'm absolutely delighted with that and hopefully I can lead the team out at Selhurst Park."

This comes after the country's highest court ruled earlier this year that Mr Zuma repay some of the $23m of public funds spent on his house in 2009.
The upgrades included an amphitheatre, pool, chicken run and cattle enclosure.
Mr Zuma must now repay the money - about 3% of the total spent - within 45 days.
An anti-corruption body, known as the public protector, ruled in 2014 that Mr Zuma had "unduly benefited" from the non-security renovations to his rural home in Nkandla in South Africa's KwaZulu-Natal province.
In March, the Constitutional Court then ruled that he had violated the constitution when he failed to repay some of the money.
It gave the treasury two months to come up with a figure for Mr Zuma to repay.
The treasury says it hired two independent quantity surveyors to conduct separate investigations to come up with the figure.
The main opposition Democratic Alliance (DA), one of the parties which brought the case, welcomed the treasury's report but said it had hoped the figure would be higher.
"This sends out a clear message to those involved in corruption, especially those in the ANC, that you will be held accountable for your actions, even if you are the president," the AFP news agency quotes DA spokesman Mabine Seabe as saying.
What has been a long and drawn out fight between President Zuma and opposition parties seems to be coming to an end, reports the BBC's Pumza Fihlani in Johannesburg.
The Nkandla scandal has been a hotly debated subject in parliament for almost three years and has damaged the president's reputation, she says.
In April, President Zuma apologised for the controversy and said he would abide by the Constitutional Court's ruling.
A few days later he survived an impeachment vote in parliament, showing his opponents that he is a true political survivor, our correspondent says.
What Mr Zuma's money will go towards
How Zuma's Nkandla home has grown
Profile: Jacob Zuma
Premier League newcomers Huddersfield Town are "more than happy with being written off", according to captain Tommy Smith.
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South Africa's treasury has recommended President Jacob Zuma pay back $509,000 (Â£385,000) to the government for upgrades made to his private home.