New Zealand is one of the countries first in line for an agreement with the EU and Smith said Trump's view on the white paper proposal was a "global concern".
"What one's got to sift through is what he did say that actually made some sense and the issue is we need the UK to become a global player.
"Trump has pulled the US out of being the leader, say, on global trade. And we need the UK to step up."
He said it was questionable whether the White Paper made that possible.
"A half-way house is not going to work. They've got to see that they've got to actually leave the EU and negotiate a real world-leading FTA with the EU."
Smith was High Commissioner to London until 2017 and is a former Trade Minister.
He is now on the influential Legatum Institute's trade advisory group which is advocating for a "hard Brexit" when it comes to trade.
He had met with former Foreign Minister Boris Johnson a few weeks ago before Johnson quit in protest at the White Paper proposal.
"A lot of these people understand what's really needed for the UK to become Global Britain, if you like, again, so they can actually join the Trans Pacific Partnership, negotiate FTAs with Australia, New Zealand, Canada, the US. A lot of these guys understand that, but the dilemma is how do they get there?
"And this paper is pretty much a compromise."
He was sceptical about whether the EU would agree to the White Paper, saying elements of it looked as if the UK were trying to cherry pick the best bits from the old EU arrangements.
The White Paper proposes a free trade area for goods, with a combined customs territory for under which the relevant EU or UK tariffs would be applied depending on the destination of the goods. However, the UK would stick to the EU's rules on standards for goods and agriculture when it came to standards.
That is seen as impinging on its ability to make its own regulations in those areas.
Britain's Trade Secretary Liam Fox has said critics had misunderstood the degree of freedom the White Paper would give the UK in free trade talks. It will have complete freedom over tariffs and quotas, enabling it to make concessions needed in free trade talks. Speaking on BBC Radio, he downplayed the impact of the common rules book, saying the UK had no intention of reducing the standards for those goods covered in any case.
Smith said Trump's comments praising former Foreign Minister Boris Johnson and saying he would be a great Prime Minister on the eve of his meeting with May were "unhelpful" but other countries needed to ensure Trump's rhetoric did not derail the trade system completely.
"It is a really tricky time here and there's no doubt that President Trump's comments haven't helped. To be endorsing Boris at a time where the Prime Minister is facing some challenges ain't easy. It ain't helpful."
He said it was Trump's style to "go over the top" and he had ruffled feathers in the UK and at NATO.
"But we must make sure that he doesn't let us fall into the trap of getting too shaken up by it all."
The Legatum Institute was founded by Christian Chandler and advocates a "hard" Brexit. It is regarded as having influence over the British Government – Smith replaced fellow New Zealander Crawford Falconer in the group after Falconer was appointed the chief trade negotiations adviser for the British Government.
