In the wet season (November - March), Broome's population decreases by about two thirds, and although it is much hotter and wetter, there is still enough going on to warrant a stopover. Just avoid swimming at the beaches during these months as the jellyfish are in. Hitching to and from Broome in the wet season is not as easy as in the dry, but still very doable.
You can probably get a ride from the middle of town if you tried, however if you head to the end of Old Broome Rd to the junction at Gubinge Rd (See the map for exact spot), you will get double the amount of traffic. Make sure to start early (5 or 6am!) if you want to get long distances! You will possibly offered a ride out to the Roebuck Roadhouse 30km from town- take it, as this is a good spot to ask truckies and travelers for rides north or south.
North to Darwin
Don't listen to Wolfe Creek!! If you get a ride with a truckie, it is likely they will take you all the way to Darwin (if they are heading to Wyndham, which many are, tell them you are trying to get to Darwin, and they will drop you at the Kununurra turnoff).
South to Perth
Again, the Roebuck roadhouse is a good place to start (unless you get lucky in town.) It is at the turnoff to the highway heading south, so plenty of people are heading that way.