The first referendum was very limited in its scope. It asked people if they wanted to leave but didn’t specify how people wanted to leave.
Holding a second referendum which gave people the choice between a hard, no deal Brexit, and Theresa May’s plan would allow people to choose the terms over which they wanted Brexit to occur. If the public wants to prioritise escaping from EU jurisdiction and the immediate control over our immigration policies, they can vote for a hard, no-deal Brexit. If they would rather remain in the customs union and avoid the imposition of tariffs, the introduction of extensive bureaucracy and customs declarations on all exports to Europe, and the uncertainty over Britain’s future trade with the EU, they can vote for Theresa May’s plan, which includes a backstop for Northern Ireland and a transition period which would see Britain remain in the customs union. The second referendum wouldn't undermine Britain's democracy as the scope is sufficiently different from the first. It would just add clarity to what the people want.
This option refuses to acknowledge the glaring issues with the first referendum. Not only did the Vote Leave campaign break campaign laws, but high-profile Brexiteers lied extensively. As a result, the first in/out vote was not legitimate. We cannot move onto the 'how' until we have clarified the decision in binary in/out terms. It also doesn't account for shifting public opinion. Given polling now indicates more people would prefer to remain in the EU than accept the terms on offer in Theresa May's deal or invoke a hard Brexit, the original question should be revisited.
The first referendum only dealt with the issue in binary terms. The government should consult the people for guidance over ‘how’.
Rejecting the premises
The first election was illegitimate and didn't deal with the issue at all. We can't move on to 'how' until we have clarified the in/out vote.