To clarify: There's nothing wrong
with playing 1 finger per fret. I have relatively small hands (about 7" from base of palm to tip of middle finger) yet I can stretch 1-2-3-4 on the bass no problem, even at the 1st fret. (Of course, I've also been playing a long time.)
However, this technique is usually totally unnecessary, and possibly even harmful (especially for a beginner). At best it simply expends unnecessary energy; at worst it can lead to cramps, tendinitis, etc. because keeping the hand fixed in a stretched position (rather than naturally allowing it to expand and contract, stimulating blood flow) is very fatiguing. (Think about when you are carrying shopping bags to the car, how good it feels to put them down and wiggle your fingers to get the circulation back.) What I recommend instead is to keep a totally relaxed, unstretched left hand (1-2-4 fingering) most of the time, and only stretch out when it's necessary for a particular lick, and then relax back to the "neutral" position.
I just stumbled upon this video. If you watch bassist Paul Jackson's left hand, you'll see he is totally
relaxed and effortless, even while he's playing some seriously funky bass: http://www.youtube.com/watch?v=puM38Xgz38w
Here's another good example from the Jaco Pastorius instructional video. Jaco was double-jointed and could stretch like 6 or 7 frets, but here you'll see he uses 1-2-4 fingering most of the time, and only stretches to 1-finger-per-fret if it's necessary for a particular fill/lick (and then goes back to relaxed left hand): http://www.youtube.com/watch?v=lwhkPSEXs1Q
A side benefit of relaxed 1-2-4 hand is that it really helps with the left-hand muting, which is an important advanced beginner/early intermediate technical skill you hopefully will be learning about in your future lessons with whichever teacher you end up studying with.