Look for hubs first

A node with five or six edges attached to it is a hub. If a hub is in the wrong place, it creates crossings everywhere at once. Fixing a hub early often clears half the board in a few swaps.

Red lines point to the problem, not the solution

It's tempting to focus on a crossing and ask "how do I fix this?" But a crossing is a symptom. Ask instead: which of these two nodes is further from where it should be?

Count crossings before and after

Before committing to a swap, mentally estimate how many crossings it creates versus resolves. Any swap that reduces the crossing count by two or more is a strong move.

The outside nodes settle last

Nodes on the perimeter of the solved layout tend to have fewer connections and are easier to place at the end. Focus on the dense centre first.