Will changing clothes last minute help me get more souls?

I'm mostly concerned about bosses, but I guess this could apply to other enemies too. Upon killing most bosses, they usually have some dramatic death animation, and only when it's totally finished will you be rewarded with whatever phrase replaced "YOU DEFEATED", the boss's soul item, and a bounty of souls to level up, buy stuff, or promptly lose due to some careless mistakes.

Between the time I land the last blow on a boss and the payday, I can usually navigate through the equipment menu and put on the Symbol of Avarice. Will this actually improve the amount of souls I get if I get it on before the souls appear on my screen? Or is that calculated as soon as I land the killing blow?

Yes, it will.

I personally tested it. You could do it as well - just equip soul gain boosting item and compare gained amount of souls with wiki for that boss.

Items, that boost soul gain are:

• Covetous Silver Serpent Ring,
• Symbol of Avarice,
• Shield of Want,
• Medicant's Staff.