Yes, it did. It changed slightly over time, but alphabetical order was certainly a concept. We even have a category of ancient texts which consist solely of the alphabet, called "abecedaries". Amnon Ben-Tor in his "Archaeology of Ancient Israel" states that they may have been writing exercises by students or scribes, but they may have additional meaning as magical or religious symbols.

The Zayit Stone is one of the earliest such inscriptions, dating from the 10th century BC, and which generally follows the modern order of the letters, with a couple of exceptions (a number of letters are switched, such as vav and he, chet and tet, kaph and lamed, etc).

Ben-Tor gives a few additional examples in the link above, if you want some more examples.

The Izbet Sartah inscription is another extremely early one, perhaps a bit earlier than the Zayit Stone. Again, it varies from the modern order with a few letters out of place (Nun is before Vav, zayin/chet backwards, ayin/pe backwards, etc), but generally corresponds with the other inscriptions we have.

We know they soon established an order and used it, because later on they used a literary device known as acrostic, which means that each successive verse in the poem begins with the next letter of the alphabet. Several of the Psalms in the Hebrew Bible were composed in this way, including the longest, Psalm 119.

Additionally, if numbers were not spelled out (such as two, three, two hundred twenty instead of 2, 3, 220), letters were sometimes used as numeral signifiers, each letter holding a numerical value as well as a phonetic value. This would have strongly depended on having an established order to the alphabet.