"Margin is closing on these votes & we will keep at it", tweeted Keir Starmer, Brexit spokesman for the opposition Labour Party, which favours a customs tie with the EU.
May, vulnerable in parliament after losing her party's majority at an ill-judged election a year ago, has come under fire from both wings of her party over a hard-won Brexit plan, with one ex-minister calling it the "worst of all worlds".
But despite the rebels' rejection of this overture, Mrs May emerged triumphant by a margin of just six votes, as the Commons rejected the key amendment by 307-301. Hammond told MPs: "This does not undermine the bill, it keeps it on the road".
The comments came as MPs resumed debate of the Trade Bill, which is one of a series of "Brexit Bills" that intend to adjust United Kingdom legislation in preparation for when Britain leaves the EU.
But Downing Street, which agreed earlier to accept the four amendments, said they were "consistent" with the White Paper where it sets out how it wants to trade with the European Union in years to come.
Theresa May has gone down to defeat in the House of Commons over her Brexit plans, as MPs voted for continued United Kingdom involvement in the EU's regulatory system for medicines.
Mr Rees-Mogg said the amendments were "broadly in line" with government policy, which is why the government accepted them.
One of the amendments practically says that the United Kingdom should not be collecting tariffs for Brussels unless the European Union does the same for Britain, one of the key ideas in May's original Brexit white paper that was created to keep the Irish border invisible.
On two of those votes, her majority was cut to three.
The government lost another, less crucial vote on another backbench amendment calling for future participation in the European medicines regulatory network.
The votes were preceded by frantic activity in the Commons, with live negotiations conducted on the floor of the chamber as Government whips led by Julian Smith made approaches to rebel MPs.
But the string of concessions infuriated Conservative Remainers who refused to back the amendments to the legislation.
The Father of the House said: "The moment that we have an important Bill like yesterday and today, the government is anxious that the House of Commons should have no opportunity to talk about it, limited opportunities to vote about it and it should be got out of the way as quickly as possible and I really do think the convention needs to be challenged". It said it objected to the amendment because it requires the government to take "all necessary steps" to join, while the government wants to join only if it can negotiate reasonable terms.