CW,
I just noticed your question to me under the "see your handi" thread. So with my morning coffee I will give you the long version as it may help you down the road.
Short answer - no it didn't just shoot like that from the start but was close yet less consistant.
I picked up this Handi (my first) last late summer and have fiddled with it likely more than any other rifle I own. Likely why it is one of my favorites now. Keep in mind it was my first Handi, so I was a rookie with them. Here is what seemed to help.
The pretty yellowish "Hornet" stock I added had nothing to do with accuracy gains in fact it hurt at first. But, because of this stock switch and help from "Q" and others here I finally got it "about lock up". First, I think the barrels them selves are fine, nothing wrong with them and I like the faster twist 1/12 in the Handi Hornet along with the single shots ability to shoot longer higher BC bullets if one wants. Between the two stocks I had 1-2 "O"rings on and off, foam pressure points at the end off the forearm on and off, swaped stocks on and off, etc.
Here is exactly the things that I feel actually improved accuracy and more important consistent accuracy,poi,etc. in no particular order.
* lock up via forearm length. The little difference went a long ways. The original wood stock was fine. The yellow laminate forearm came a bit longer length. It was to tight and did not shoot as well. I took it down a little, it was then to loose and did not shoot as well as the original wood. Finally, I realized if I wanted to use this laminate I bought I have to get the lock up right. So, I in this case had to shim the shoe with aluminum and what worked really great for me was I epoxy bedded the shoe to the forend as such. This filled small gaps and kept things square. I am a fly tier, flyfisherman and I just used 5 minute expoxy. My lock up is such that from a horizontal position while holding the butt with one arm and breaking the action with my thumb the barrel will fall to the ejector click point but it will not "click" the ejector........extractor.......what ever. It does not just loosely fall open. From that point I have to pop it with my left hand to eject.
* I sanded out the forearm channel with a dowel from the lug forward and it wears a "O" ring. Basically, with the twon adjustments it is free floated from the lug forward and bedded if you will, at least in break action terms. If I put the rifle in vertical position butt down, there is no side ways free play in the end of the barrel at the forearm tip. It won't push from side to side. This is because of the lock up. From there my forearm screw is tightened down well plus 1/4 turned and marked with a dot from a awl.
* Length of pull, I put a butt plate on to lessen the L.O.P. about 1/2". My scope 4-12 Banner can't go back further and it seemed I was stretching to get full view always. It does not have long eye relief. This combined with the LOP being to long for me, put me in a maxed position of reach and cheating over the top. The butt plate solved that, everything is comfortable and I have full scope view naturally. For me, I shoot best when I am comfortable.
* Reloads- I tried a few yet just use a simple Lee Classic Loader, plastic mallet and a 12.8 grain Lil Gun a very avid shooter pal out west made for me and mailed it to me. When I got to the 40 grain V Max it shrunk a bit more. Particularly at distance. I tried various O.A.L. of the reloads and it made little differance. I now let the bullet jump to the rifleing. I think the simple Lee Loader works for me in the Hornet because they are once fired ammo in my Hornet chamber and then neck resized in the loader. I do clean the primer pockets out well and chamfer

? the inside of the neck, with the additional gadgets. I am using federal and CCI small rifle primers. It does not seem to matter much which.
* Dry- I keep the action bone dry free of any oil & solvent. This is a time saver anyways. My Hornet only powder fouls, I never was able to get a trace of copper from it despite using some of the best copper removal products available. On the negative side, solvents and cleaning the bore with rod, just left oils in the action which caused problems. I now just run a dry boresnale through the barrel after woodchuck hunting,range shooting,etc. to remove powder and keep it in the groove. I only clean with chemicals as needed, which appears to be a long time. If I do, I get the action as dry as I can after and continue wiping after shooting.
* Tight Grip- With this rifle I now use a tight grip pulled well into my shoulder for all shooting. Off bags I keep one finger widths distance between the front sling stud and the bag itself. The floating bedding allowed for consistant POI doing such. As an avid woodchuck hunter whom normally shoots from sitting position using shooting sticks. I have tested my P.O.I. with sticks as compared to bench and bags several times at the range and am convinced it's now the same or of little differance with the above float & bed set up.
* The above is what helped with consistent accuracy as compared to spotty. If I picked one that stood out the most it would be lock up. I don't have the gunsmith tools,mic's nor experience that some here do. This has just been my experience. I now know by feel how I want that lock up and the description above is it.
* My trigger is still as is from the factory. Mine was never bad. Did not measure it but from experience I would say it's about 3.5 pounds. The shoe is pretty wide which helps and as long as I squeeze it rather than pull it, it's fine. I do pull all the way through as noted in the tips above. I may tweak it this winter but, the trigger as is from the factory in my case was better than many factory guns costing more.
Probably if you asked me six months ago if "I would ever buy another Handi Rifle?" my reply would have been "not likely". That would not be the case today in fact I may get a .243 Why? Because I now understand them and they are very simple to tweak and I think they have good barrels. This makes them a good value and I am convinced they are as good as break actions costing many times more. In Hornet I would put them up against all except the CZ,Annie,etc..
If I bought another I would do the above before I even shot it with various loads or ammo, and yea I would buy another with a laminate stock............I like the camo version amongst others.

Shared just to help. Not a gunsmith just a varmint hunter. Use cafeteria style, nothing is in stone
