One of the best ways to improve your skill at reproducing notes is to develop your ability to sight-sing, which is the ability to read music at sight without the help of an external instrument. This primarily involved training your "inner ear" in a process called audiation. Audiation is essentially hearing something in your head before you externally reproduce it - like the way you might have thoughts said to yourself (like J.D. on the show "Scrubs" when he had an inner-monologue.)
The difference here is that instead of words, you are training your brain to "hear" the pitches first before you sing them. Once you "hear" the pitch in your head, then you externalize it by singing.
The process of transcribing what you hear in your head is a skill that composers, jazz musicians, and most musicians use every day.
To develop this skill, I would recommend purchasing a sight-singing resource, starting on page one, learning about fixed do and moveable do, and then perfecting every study in the book cover to cover. If that seems a little daunting at first, you could expand on what you're currently doing at the piano.
In addition to singing perfect-fifths, you could sing other intervals as well, or test yourself to see how well you can reproduce scales. You can make up patterns and see how accurate you are in reproducing them. This brings me to my next point:
Developing your pitch memory can go a long way toward helping you remember the pitches you've already sang, which you can use as references to help you find your next pitches. This can be done in ways I alluded to above, or you can find little software programs online referred to as "pitch memory games" that can be quite fun.
These suggestions are by no means comprehensive, but I just wanted to get the ball rolling.
Hope that helps.