There is many types of bluing which one do you intend to use? Basic answer is adding onto existing bluing with say a cold blue will seldom be even in colour all over .
It's really a patch up system . Sometimes you can be lucky and it matches in ok and other times not so good. This is because there is many bluing processes and they all come out slightly different in colour and texture and oil and grime on the metal also changes things . To get the best outcome you do need to strip all the old blue off , polish up the metal all over , degrease the surface properly and then re-blue . The most common process used by smiths is hot caustic bluing . The polished and degreased metalwork is suspended in a vat of hot caustic salts for a short time and it sends the metal blue. Then it's washed and should be boiled in plain water to remove salts and then dipped in oil . Keeping the salts out of the bore is also a good idea but most don't bother .
Cold bluing can be much better if the metalwork is stripped back and polished up and degreased , this video gives an idea
https://www.youtube.com/watch?v=X-CDOJF-TmY However to strip and polish ( buff) metalwork properly you really need to be able to remove the barrel . However it's not impossible with the barrel and action still attached just more difficult . Then you have London rust bluing which is basically once the polishing and degreasing is done you rub a rusting agent all over the metal and let it sit for 24 hours . This causes fine rust to start then you boil the metalwork in water to convert the rust to a stable oxide and then let it dry. Then polish off the black oxide with fine steel wool and do it all over again and again until a deep blue appears.
There is also several chemical formulas that can be used in a hot blueing tank other than Caustic .