As the second segment is the most limiting, using more flaps for takeoff won't help, it will in actual fact decrese the net climb gradient...however it might well decrease the takeoff run distance.
Now, if you have available lower certificated flap settings for takeoff, and indeed the runway is long enough, this works as well.
Also, if a Boeing aircraft, why not use the 'improved climb' data provided.
Many times benefits can be achieved here as well.
Lockheed called this 'overspeed'...same results*
More power?
Ah...yes, this will work.
But, considering the fact that you might be using this already, the next step would be to limit payload...not a happy thought for the marketing department, however sometimes necessary, without requiring a tech stop for refueling.
Occasionally, the tech stop is more economical, it just 'depends'....
Many variables involved.
Then we come to re-release, and in doing so, oftentimes the fuel uplift can be minimised, thereby eliminating the (otherwise) required tech stop, weather permitting, of course.
Quite a few ways to skin the performance cat...!
*In prior years (B707), Boeing called this overspeed as well, but changed to 'improved climb' some time later.