You can move the Major Image only by using your action. There are no other ways to manipulate the location of an illusion, so that's your only option.
However, normally this is fine. You can travel with your illusion by constantly giving your attention to it and making it travel with you. Any time you need to do anything complicated enough to require your action the illusion will stop though, so you're going to be slightly burdened. Although it no longer requires concentration, you effectively still need to “concentrate” most of your attention on it.
Multiple Major Images
And of course, you can only have an Image travel smoothly with you with one Major Image at a time, because you only have one action at a time to use to move one. To have each of two Images move continuously isn't possible, but you can move slowly and spend alternating actions on moving them: one would move, then stop, then the next would move, then stop. Unless, of course, you're OK ignoring the Images for a bit and having them suddenly jump from one place in the 120-foot range to another — that would let you move at full speed, and keep them with you.
You'd look like you had an entourage of flickering entities following you across the countryside and through towns, which would be uncanny but might be an impression you'd enjoy. On the other hand, that unnaturalness would likely give viewers saves to notice that they're all just illusions, so that's a downside and would mean that anyone observing you might be “vaccinated” against your come-along illusions. (They might even get easier saves against all illusions when you're around, since they'll know that other weird or unexpected stuff happening near you could be just an illusion.)
Basically, one Major Image is doable with full animation, so long as you're giving it all your attention that isn't needed for movement, speech, and simple object interactions. Two is slower, and won't result in full animation. More is possible depending on how quickly you walk (because their continued existence is limited by the 120-foot range of the Images from you), but they will be flickering around and look static between jumps.
And you still need all your non-trivial attention to bring them along with you.
Sleep, unconsciousness, and incapacity
Another wrinkle is that you'd have to make sure you never slept or were otherwise unconscious while travelling, such as in the back of a wagon or napping in the saddle, otherwise the images would quickly fall more than 120 feet behind and you'd stop being able to change their location again until you can backtrack into range again to “pick them back up”.
Also, don't get abducted in your sleep, or fall down a cliff, or fall unconscious and need to be carried to safety, or suffer any circumstance that would prevent you from using your action to keep your Image(s) with you and moving. (Those are worth avoiding on their own, though.)
Extradimensional items don't help
You can't circumvent this with a Bag of Holding or a Portable Hole either: the spell specifies that you can move the Major Image to a location in range, and the inside of a Bag or Hole are on different planes of existence (they're pocket dimensions), which is out of the 120-foot range. You can't get around that by moving the illusion to the “entrance” of the item either: it's not an object and won't be passed through the entrance by the item's magic.
In conclusion, it's probably not worth it
Having to spend all your active attention on keeping one or more Major Images with you, is probably more work than it's worth. You might save a few spell slots, but being that inattentive while travelling will probably make life slightly more difficult, possibly even causing disadvantage on Perception checks and other circumstances where having your attention be complete free is expected.
And your come-along illusion(s) would be slightly less useful than a normal, freshly-cast one, because if you want to “recycle” it suddenly there are now two chances for a victim to notice something odd and get a save: not just the appearance of the effect you want them to see, but also the sudden disappearance of whatever your come-along illusion looked like a moment before. It adds a vulnerability that makes the come-along Major Image slightly more situational than a fresh one normally would be. And that's assuming the intended victim hasn't already seen your Major Images doing weird things (stuttering and jumping around) and already made their save to know it's just an illusion.
Basically, you'd be distracted all the time, for the benefit of saving a few spell slots and having less-useful-than-normal recyclable Major Images following you around everywhere.