Bigger hands can be achieved by using the "winds" and extra points for symbol sets.
The way I learned to play, is you need either 4 sets and a pair, or 4 runs of 3 and a pair, or you can mix and match, plus a pair. However, if you mix and match then you aren't able to take the discard of other players for the winning tile, and have to pick it up yourself through the stub.
Getting a set of symbols, the green, red and blue, gets you 1 extra point. Getting 4 of a kind too. If you get a set of the "direction of the round", or your "direction" based on your seat relative to the dealer, then you also get extra points.
Then there's the holy grail, royal flush hand which I forget how much it pays but it's a huge amount. 1 & 9 of each suit, 1 of each symbol and a pair.
Some mahjong sets come with "birds" which get an extra point just by getting a single one which you immediately place face up and get another tile from the back to replace it, similar to quads. I'm not sure but I heard it's the Taiwanese version? I've never learned official rules, this is how I've always played with family and when I've played with others, the rules are the same around here.
If you're already playing by these, then I don't know if there's any other variations that would lead to more gambool.