All woodwind instruments1 have a first octave, which then overblows to a second octave. Then, higher notes are possible through increasingly convoluted fingerings. So you don't have to do much to assure that the instrument will have more than one octave. Modern instruments have octave keys and other mechanisms that make this easier, but even the most basic instruments are capable of over two octaves.
Here is a fingering chart for fife and tin whistle. Look at the fingerings for notes in the D major scale, and ignore the others for now. Notice that it's one finger/hole per note. It would be great if we had 12 fingers, then we could simply have one finger/hole per note in the full chromatic scale, but alas we don't. So to get the notes in between, we have to use some trickery.
On the tin whistle/fife chart, you can see that most of the in-between notes are achieved by half hole fingerings. This is exactly what it sounds like--you cover the hole part way, and only some of the air can get out there, and some has to continue down to the next hole, so you get a pitch in between. This is as awkward and imprecise as it sounds.
Now consider recorder fingerings, especially the top half of the first octave. The finger holes on a recorder are much smaller, so it's as if the half-hole effect is present on every note. Even on a normal fingering, not all of the air gets out the first open hole, and we can exploit this in our fingerings. By fully covering holes after the first open tone hole, we can very precisely control the effective length of the instrument. Bending a pitch down in this manner is called shading, and a fingering that uses shading is called a cross fingering. So the in-between fingerings are much more precise, but they're still awkward. And now it's hard to figure out exactly where to put each hole, as every hole's placement affects every note above it.
So you have to decide which of these paradigms to use. Your instrument will either be fife-like, with big holes and half-hole fingerings, or recorder-like, with small holes and cross fingerings.
Either way though, it's going to be hard to play in any key. It will be easy enough to play in the scale that the instrument is designed around (the one where you just add or remove one finger at a time), and not bad in closely related keys, but playing in a distant key will involve some ugly transitions between notes.
Modern instruments have alleviated these issues with complex mechanisms. They do have one hole for every note (sometimes multiples, for alternate fingerings), and various holes are covered automatically when certain keys are pressed. One of the first improvements was made by Theobald Boehm for the flute, which was then used for clarinet and saxophone. With this system, the fingering that was F♯ for fife/tin whistle/recorder is now F♮, and between F and G there is an extra hole. Instead of covering it with a finger, a mechanism automatically closes it whenever any of the right hand keys are pressed2. So to play F♯, we leave the F hole open but press one of the lower keys in the right hand. Thus the F♯ pad is closed, but the F hole is open and F♯ sounds. Saxophone and flute use a similar mechanism for C♯ vs. C at the top of each octave.
1 Except clarinet. Clarinet operates on the same idea, but it only sounds its odd overtones, so the gap between the first "octave" and the second is actually a 12th.
2 Saxophone and flute use keys instead of covering tone holes directly. Clarinets do have holes that are covered directly, so modern instruments have rings that get pressed when you cover a hole.
