I can't speak anything for the GBA because I never coded on it but on the SNES, I certainly know quite a few tricks which allowed the devs to squeeze out more colours from a background.
The biggest one is that the background colour can be used as an additional colour which is extremely important for 2bpp backgrounds given they can't use too many colours on a single tile. In some cases, the results can be rather unexpected given what is the background colour from a human perspective can be different that what is the backdrop from the SNES's perspective.
Further complications can make HDMA. Many games use it either for Mode 7 magic, parallax scrolling or a colour gradient in the background. Said background colour can be blended (more specifically, added or subtracted) together with a background as seen with the
the Spinning Tower in Super Castlevania (not necessarily an example on its own but it is a possibility).
Others write to the palette instead and add in more colours that way. The background of 3-4 in the SMW hack A Plumber for All Seasons is one such example where the darker tree stems are on layer 3 and the brighter background on layer 2 even though it looks like there are too many colours for layer 3 to handle.
And the sunset from 3-1 and 6-1 in Yoshi's Island is an example which combines the backdrop trick with HDMA where the part of the blue/red of the clouds has to be moved to the gradient because the ground (the green/brown part of the background) already uses three (unique) colours, same spiel with sun.
Admittingly, using these tricks really blur the line to what's possible on the SNES or not (the aforementioned forest background is the biggest offender) so it's generally better to say that these tricks should be used more as a last resort or if you know what you're doing, especially since the amount of HDMA channels is limited and colour maths also comes with its own limitations as well.