For general nutrition as well as relief from constipation, I would think you would want to introduce more whole grains - whole wheat pasta, brown rice, and so on. White rice is something that is part of the "BRAT" diet when they have diarrhea - "bananas, rice, applesauce, toast". Why not try mixing half brown rice and half white rice first, then gradually switch to all brown? I save the water from steamed vegetables, and use that to make rice - start the brown rice since it takes longer and then add the white rice about halfway through the cooking time. Start with the total amount of water you will need even though it's twice what the brown rice will absorb - if you add more water and the white rice halfway thru, it will stop the cooking until the water boils.
Since she hasn't had pasta yet, I would start with the whole wheat. I think the curlier shapes are easier for them to pick up so try rotini or anything with lines on it - I think it's called "regate" - like penne regate or ziti regate.
Have you tried whole wheat bread? Again - the fiber will prevent constipation. I use it for French Toast and add extra protein powder to the egg batter. I add cinnamon to the batter - it's good for you and it gives a nice flavor, reducing the need for a lot of syrup. Added nutrition will help everyone, especially a small baby.
Just make the switches slowly - don't go immediately to all whole grain if she isn't used to it. Let her system gear up for it! But if you never introduce white flour and white grains, you don't have to then switch them off it!