Bryant became the first player in history to go 5-for-5 with three home runs and two doubles as the Cubs beat the Reds 11-8.
"Maybe when the bat breaks, I'll send that," Bryant said Tuesday. "I'm keeping the jersey."
Bryant became the first player in at least 100 years to hit three home runs and play three different positions in a game, according to the Elias Sports Bureau. He started at third base, moved to right field, then ended the game in left.
"It's [moving around] not hard for me because I've been doing it for most of the year," Bryant said. "It doesn't matter what I've done that half inning [on defense]. I turn on a different switch."
Bryant is the first Cubs player with five extra-base hits in a game since 1885. He has 12 home runs on the road already this season, after hitting only five last season. The leading vote-getter among NL third baseman for the All-Star game is hitting .278 with 21 home runs and 57 RBIs.