I don't know exactly what gauge your pattern asks for, but let's just use an example of 20sts/4". If that produces a hat for a 0-6 month old using size 7 needle, say you tried a swatch with a size 8 needle and you got 18sts/4". That would work out to 4.5sts per 1", so you could take your total stitch count and divide it by 4.5 and you would get your approximate finished size in inches. Say your cast on stitches total 60. With your original gauge (20sts/4", or 5sts/1") your hat would be 12" (60sts divided by 5sts/1"), and if you used the gauge of 18sts/4", or 4.5sts/1", your hat would be 13.3".
Again, this is just an example, I don't know what your stitch counts are, but you see how much difference even 2sts/1" can make! Over 1" difference in finished size. You also have to remember that a knitted item is fairly stretchy so you might not need to alter it all that much. I'd try going up one needle size, as we discussed in this example, and see if you arrive at the result you're looking for.
Let us know if we can be of further help.