Keep the PL low, and concentrate on the Combat Advantages (née Feats) listed on page 70. That's where you find elements analogous to HERO's maneuvers. A practitioner of a "hard style" might use Accurate Attack, Improved Critical, and Takedown; someone using a grappling style could use Grabbing Finesse, Improved Hold, and Prone Fighting.
When I read this I thought: Just don't.
In "the real world" there is so much more random stuff that happens than meaningful stuff towards some goal that its the decision of the people on place to decide which dungeon to crawl and which to ignore.
If one wants to explore every single house in every street then in how many of that houses you will find something worthwhile? In the first hundred or so you find lots of useful stuff, good loot interesting challenges but then it starts to get boring and you start to skip houses you think are not worthwhile.
On the other hand there are houses that are interesting far longer because they are far rarer. An industrial complex vast enough to hold its own population that never needs to leave it holds buildings that are super interesting all the time.
As long as you give them enough information on how the hexes look like / what to expect from the NCSs (through rolls against social skills) the players will soon start to skip stuff and start to learn when something is "just random" and what stuff is "significant". (And think about all the interesting plot stuff that happens if they get it wrong.)
This learning process is one of the things that makes sandboxed games so interesting.
Also, sometimes people can't skip random stuff. That happens, really. The gang that wants to mop them up is going to pray on them, either because they carry around significant amounts of cool loot, or they happen to have the wrong shade of gray skin.
After about 10 sessions or so (your call) you should allow your players to make one roll if they think their current encounter is significant to their personal plot to fasten things up.
If you go this route you should tell your players so and recommend to them to keep notes, because it happens that they skip stuff they should not have skipped and they need to be able to figure this out afterwards.
As a sidenote: I found it very hard to come up with "good" randomness in such situations. I use the traveller world generation stuff for this, only I generate "countries" (in my Post-Apoc setting more like "tribes" or "warlords" but its all the same) with it. Works like a charm. I generate the countries by the Traveller rules and then write two to five lines on it what it means in the world. Stuff that just won't fit gets re-rolled.
Best Answer
Superheroes are reactive because the default state is that they've won: at the end of each episode, they get the city back to the way they want it, and there's nothing more for them to do until something goes wrong again. To get active superheroes, you have to create a situation where the default state is the heroes are losing: something is wrong with the city and it will take the whole campaign to fix it.
You could do a war story, perhaps: our heroes have gone behind enemy lines to sabotage the enemy war effort and capture enemy superheroes. Without killing anyone, of course.
Do a crime story. The city is infested with gangs, the police aren't solving it, and now it's your job to clean it up. The Batman solution, dressing up in a costume and punching individual muggers, clearly won't work. Climbing the hierarchy and dealing with the leaders might work better. But a real solution should involve reforming the criminal justice system and addressing the socioeconomic causes of crime...
Bonus points if the crime story is actually set in Gotham City, and Batman is punching muggers in the background and wondering why it's not solving the problem.
Do an infestation story. AI (or alien parasites, or zombie virus), has infected and mind-controlled every citizen in Chicago. It's spreading, but slowly. Our heroes show up in the city and have to avoid getting infected themselves. Then they have to figure out what the infestation is doing, thwart its devious plan, figure out where it camr from, and devise a plan to cure it.