Not a 77/357, but a 77/44 so much the same.
Recently started my journey with this rifle and it will be a cast bullet shooter. "J" bullets are NOT an option.
Now, about the first thing I did was float the barrel. ASAP!
Then after my first range session, sighting in the scope and shooting my first two series of test loads. AA#9 and H110 behind a 275gr Wide Flat Nose cast I was quickly ready for some after market up grades.
My son has a 77/357, so I duplicated my order for after market parts and they are on their way to him as we speak.
Go to TriggerShims.com for shims to take some play out of the trigger, a shim kit to remove excess play in the two part bolt. NOTE! there MUST be some play between the two sections to allow the bolt to close.
Then the most worthwhile of the upgrades was the Volquartsen Target trigger sear.
Where the factory trigger pull was some where close to 2 - 2 1/2 tons, only slight stretch here as it was VERY heavy, it is now light and crisp. Awesome in fact!
That trigger sear is also available from Lance at TriggerShims.com.
Am planning on a second range session next Wed.
If you can find a copy of RIFLE #263, read the article by Brian Pearce on the 77/357 and 77/44.
I hear about it, and a nice forum person sent me a CD with the article.
Crusty Deary Ol'Coot