TAP Portugal has a weekly flight from Miami back to Lisbon that leaves at the fun time of 03:45.
Of course, whether it is the 'worst' time is subjective and will vary from person to person.
A handful of other US-Europe routes have flights that depart at 01:00 or later.
This year, more scheduled passenger flights will depart from the US for Europe at 18:30 (local) than at any other time. It is a sensible and normal time of the day. However, using Cirium data to examine all planned departures on all routes shows that a very 'different' time is available.
Fancy leaving for Europe at 03:45?
Between July and October, one of TAP Portugal's 10 weekly flights from Miami back to Lisbon will leave at 03:45. The first date to see it is July 4, with a Thursday service until October 24. It follows a 03:25 departure that will run on Thursdays between April 4 and June 27. It is a massive ouch if you are a tourist and must leave your hotel numerous hours earlier.
Using the 298-seat Airbus A330-900, the 4,157-mile (6,689 km) service is bookable. The full schedule is as follows, with all times local:

Lisbon-Miami: TP225, 20:20-00:20+1 (9h 00m) ← Wednesdays
Miami-Lisbon: TP226, 03:45-17:05 (8h 25m) ← Thursdays