I agree with the Java Moss and Java fern suggestion... Usually I see Java Moss in the stores but sometimes it's actualy Java fern... theres not a huge difference between the two. I LOVE Banana plants... (thats just what they are called) ... they have a bulb that sits above the substrate that resembles a bunch of green bananas
and my Bettas seem to love them as well, as the stems grow much like a tropical water lily (they are related) with small leaf pads at the surface.. great for fish naps!
These also do not clog up the tank if you plant them moderately.
Thats the end of the list of plants that have done well for me (with minimal care).. except for those bulbs you can buy for Bettas.. the Temperate water lillies are what those packages usually contain.. and they do grow very VERY well... But too well, they clog up the tank horribly and end up not as attractive when they have grown a bit. They are temperate, not tropical, water lillies... so their leaves are a different shape (like an arrow) and they just aren't as nice a plant.
If you are going to get a much larger tank... remember with your Betta you are going to want to under-filter it. The strong currents of most filters can swish your Betta around like he is in a toilet bowl, and trust me, they don't much appreciate that. You will be able to tell his happiness level based on where he "hangs out" in the tank. A healthy and happy Betta will occupy the bottom, middle, and brief moments at the top of the water column. Bettas who hang out at the surface may be having trouble with airation and/or ammonia in the tank limiting gill function and forcing them to rely on their Labyrinth lung. Bettas who hang out too long at the bottom may be hiding out from strong upper currents caused by strong filters.
There are several good submersible filters that are adjustable you can find for your new tank. Most that hang on the back will leave your Betta holding on for dear life at the bottom. I have had good results with a Duetto filter made for 10 - 15 gallon tanks in my 20 gallon. It has an adjusting doo-hicky on it so I can change the flow level. If you end up with one of those kits with the filter already in it, or simply have limited choices at the pet store and have to use a hang-on type of filter... try putting the filter far to one end, and maybe make a little "Betta cove" with a line of taller plants.. so he can hide out from the current when he needs to.
One more stab for Fancy guppies.. hehe... If you get a trio of guppies now you will likely have a lot more of them by the time you get the bigger tank.. provided Mr. Betta doesen't gobble the fry