NZTA Auckland highway manager Brett Gliddon said those that didn't make the exit would be given a third and final warning to pull over or stop.
"It is a very clear warning that evasive action is needed, it will be very difficult to miss or ignore.
"Bridge strikes cause delays that can affect many road users, creating disruption and financial losses to drivers and businesses."
The agency was still seeking compensation from Higgins over damages to the bridge from the May accident.
NZTA's structural inspection report released to the NZ Herald under the Official Information Act showed the strike's impact caused concrete to come loose from the structure with other visible vertical cracks and distorted steel armour plates.
Truck driver Graham John Kennett, 65, has also been charged with dangerous driving over the crash. He pleaded not guilty in the Auckland District Court in June and is due to reappear on September 14.
The charge comes with a maximum disqualification period from driving of six months and is also punishable by a term of imprisonment of up to three months and a fine of up to $4500.
Since 2008 there have been 39 incidents on the Penrose overbridge, 37 per cent of the total number of strikeacross all of the bridges on Auckland's highways.
This year alone the Penrose bridge has been hit four times. The bridge, which has a clearance height of 4.57m, was built in 1952 and had its underside reinforced with steel due to the frequent bridge strikes.
The maximum height for trucks on this part of the Auckland motorway is 4.25m. An agency s pokeswoman estimated the total cost of bridge-strike damage across the whole of the city network was around $203,000 over the past seven years.
Only one strike at the Penrose overbridge, on January 29, 2015, was deemed severe enough for the agency to recoup costs of $31,120 from the company responsible, Tracta Tranz.
NZTA is launching an educational initiative for at truck drivers.