Stephen Flynn was elected group leader on Tuesday after Mr Blackford abruptly announced he was standing down.
Several MPs have since resigned from the SNP's front bench team, with Pete Wishart saying he was "bemused" about why Mr Flynn had sought a change.
But Mr Flynn, 34, told BBC Scotland that he only decided to stand for the job after Mr Blackford's resignation.
This is despite the fact he had distanced himself from the role only two weeks earlier, saying he had "no intention of standing" amid rumours that he was "on manoeuvres".
Mr Blackford, a close ally of SNP leader and First Minister Nicola Sturgeon, announced last week that he was standing down from the post he had held for five years - but will be staying on as MP for Ross, Skye and Lochaber.
This followed months of speculation about his future in the role, which included claims that Mr Flynn was seeking to replace him.
Mr Flynn then won a leadership vote against another ally of Ms Sturgeon, the Glasgow Central MP Alison Thewliss.
Allies of the new leader say he plans to set up a smaller front-bench team, which they say would allow other MPs to focus on constituency work and campaigning for independence.
And three prominent members of the front bench group have already stepped back, with defence spokesman Stewart McDonald and international development spokesman Chris Law joining Mr Wishart, who was its agriculture spokesman.
Mr Flynn described the three MPs who resigned as "incredibly valued colleagues", adding that he was putting together a "dynamic set of folk" to lead the party at Westminster.
He also pledged his loyalty to Ms Sturgeon, who he referred to as "perhaps the most effective and efficient politician in the entire of Western Europe".
He said: "Ultimately we are SNP MPs, and that means we are not concerned about which bench we sit on in Westminster, be that a front bench or a back bench.
"What we are interested in is making sure that Scottish MPs don't sit on those green benches, and that is our ultimate aim. When it comes to that myself and all of my colleagues will be at the forefront of that battle."
Political opponents have accused the SNP of "fighting like ferrets in a sack", with the Scottish Conservatives claiming that "the party's reputation for iron discipline" was disintegrating.
But MP Stewart Hosie told BBC Scotland on Tuesday - the day before Mr Wishart quit - that talk of division was "complete fiction", saying he had "no idea where these stories have come from".