One thing I would suggest and STRONGLY advise for EVERYONE...before you start banging away...is to bed the chamber in steel epoxy. Just a small bed of epoxy along both frame rails...it takes about a hour, it will center the barrel and position it the same place everytime you close it, it will take out all the "normal" slop and will make thing so much easier when you do start load development...THEN full bed the forearm groove. I've had better luck with a full bedding job on the forearm or making a hanger, installing a pillar and locking the forearm to ONE point, metal to metal, than all the other ways ...but that is so argumentative I hesitate to even mention it...it starts up a brouhaa that has no point.
90% of the Handi users AND everyday shooter don't understand that 0.005" of movement...ANY movement in ANY part of the system will cause a 1" shift in POI at 100 yds.
NO Handi I own EXCEPT those that I have bedded the frame rails returns to the same place eveytime when they are closed...and even the bedded ones aren't perfect, but DON'T flop around.
Freds way of shimming the lug also helps but still leaves gaps...I can still feel looseness with the one I shimmed and that is with the shims dragging on the frame sides. But try them, it might work for you.
I read so many accounts of people being disappointed in their groups, but I don't read too many accounts of HOW they solved their problems and some getting disgusted.
Handi barrels aren't the best in the world and temp can cause some problems, but before ANYONE can address the REAL problems of load development, the Handy HAS to have certain basic mods accomplished...or live with the limitations and stop whinning...the Handi has ALWAYS been a cheap, mediocre, questionably accruate rifle, and you CAN'T make a benchrester out of it, but you CAN make it shoot very well...it just takes work.
Luck