If you don't want to use a more concentrated developer and you don't want to increase time/temperature of the first dev, then trying a silver halide solvent would be easier than adding another bleach step (which implies using a clear bath after) after 2nd developer. This will reduce the density a little (which you want) creating a brighter image.
Once in the ballpark of the right amount to add (usually between 1 and 5g/L), you can even control the ''tone'' of the image. If the scene is contrasty add a little less and increase 1st dev time, if it is not contrasty enough add a little more and decrease 1st dev time.
That's what I've been learning since a few weeks and it works for me. Silver halides aren't necessary, but they can make things easier...