The Labour party should continue to campaign against Britain’s exit from the European Union and a public vote should be held on the terms of any replacement trade deal before article 50 is invoked, Owen Smith has said.

Smith, who is challenging Jeremy Corbyn for the leadership of the party, said a new referendum or a general election should be held to confirm the details of the post-Brexit deal before Britain begins the process of formally leaving the trading bloc.

He accused leave campaigners of lying about the benefits of leaving the EU, and reiterated his complaint that Corbyn had failed to argue strongly enough in support of the union. And he said that under his leadership Labour MPs would vote against any attempt to invoke article 50, the exit clause of the EU’s Lisbon treaty, until a second public vote is secured.

He told BBC Radio 4’s Today programme: “I’m saying that we didn’t actually know what we were voting on. Theresa May says ‘Brexit means Brexit,’ but in my view we were fibbed to about the extra money for the NHS – there is no extra £350m – [and] we were fibbed to about the suggestion there were going to be easy answers on immigration. There are not.”

Smith has made support for the EU a key plank of his Labour leadership campaign, in which there have been few other points of disagreement between him and the incumbent. Both men have advanced a broadly centre-left agenda, with policies including renationalisation of the railways, increased public spending and enhanced workers’ rights.

Corbyn has endured months of accusations from his political opponents that he failed to do enough to mobilise Labour supporters into the polling booths to vote remain.

However, Smith has faced accusations of arrogance towards voters, who voted 52-48% in favour of leaving the EU in June. Challenged over his attitude on Today, he insisted that voters had not been told the full truth and that Britain faced an uncertain future on its own.

“I think we are seeing the Bank [of England] admitting last week that we are likely to go into a recession; they have had to put an extra £60bn-worth of our money into QE [quantitative easing]; that there are going to be a quarter of a million jobs lost; that inflation is going to go up,” Smith said.

“I still agree with the experts that things are going to be really bumpy. But the crucial point that I’m trying to make … is that at the end of this, when we know what is really on offer, when Theresa May has done this negotiation … at that point we could put it back to the country, either in a second referendum or in a general election, in which under me Labour would be arguing that we should remain.

“I still fundamentally believe that we didn’t fight hard enough, that Jeremy didn’t fight hard enough. I still feel that we should stay in.”

