You could cast Deeper Darkness with the Widen Spell feat which doubles its radius, then cast DayLight without the feat at its normal radius.
I didn't think that would work because of the spells counter/dispelling each other?
It is my understanding that to dispel the opposite spell, you have to actually "target" the other spell with it*. Admittedly this area of the rules is quite bare, and seems to have been left largely to legacy understanding carried over from 2nd edition.
Counterspelling only happens when you ready an action to counterspell (special feats aside).
If the two spells simply overlap areas, then they cancel out their effects, so long as they are overlapping.
*Correction: If they are targeted spells, they must have targeted the same target; area spells must be aimed at the same area. See the second half of
this post for info from a Rules of the Game article that better defines auto-dispelling.