I have not paid attention to the 38mm bands because I have a 42mm Apple Watch (series 0), but Apple's 42mm sport bands come with
three pieces - the upper piece (traditionally goes from the top of the display over the top of your wrist and has the fastening pin on it), and
two lower pieces (that go from the bottom of the display around the lower side of your wrist, and have a row of holes for the pin). In the 42mm watch, these two lower pieces are referred to as S/M and M/L, and together these are advertised to fit wrists from 140 to 210mm (my wrist is about 180mm and fits with the S/M piece, with one or two holes to spare). The idea being that you wear the top piece with whatever lower piece works better for you.
As well, for some colors (no, I don't know which), you can also get the band wth a L/XL piece replacing the S/M piece (so the band covers M/L/XL instead of S/M/L). The sizing goes like this (copied off
Apple's page for the white Sports Band):
38mm
S/M & M/L - Fits 130–200mm wrists.
42mm
S/M & M/L - Fits 140–210mm wrists.
M/L & L/XL - Fits 160–245mm wrists.
I don't know if they offer any bands with L/XL parts in the 38mm size - Panther61 suggests the answer is no.
The third-party sport bands vary from mediocre to quite good, though they're all silicone, I've not seen any yet using the fluoroelastomer that Apple uses. Some prefer silicone, many don't mind, but others dislike the silicone ones (I think Apple's bands feel better) - so it's definitely a YMMV situation. But the third-party companies are quick to fill in the gaps in Apple's lineup - more colors, different lengths, bands in 38mm that Apple only offers in 42mm (e.g. the Leather Loop) and vice versa. You can likely find third-party 38mm straps long enough to work for you, but I wonder if the 38mm watch you tried on perhaps had the S/M lower piece fitted rather than the M/L piece - is your wrist larger than 200mm?
This may help with sizes (the lighting is bad, the color isn't quite right, but the rulers are readable):
