1957 BA race: Williams v. Mantle.

Ted Williams reversed the batting average (BA) race of 1956 by again dueling with the much younger Mickey Mantle and beating him. Highest BA after April:
Mantle: June 23 .392
Williams: May 8 .474

As late as June 5 Williams BA was .401. In 1956 Mantle was at .400 on June 8.

At the end of April Williams led .426 to .306. By June 5: Williams .401, Mantle .362. June 16 they were tied at .379. June 29 Mantle had his biggest lead, 34 points: .389 to .355. Lead changes:

Date Mantle, Williams

Jul-17

.356

.367

Jul-23

.367

.360

Jul-26

.359

.364

Aug-31

.378

.377

Sep-2

.373

.376

Williams didn't play September 2-16.

Sep-17

.366

.377

Mantle played his last game September 22.

Sep-22

.365

.383

Final game for Williams:

Sep-29

.365

.388

So Ted Williams never took a serious run at .400 in 1957 but instead battled Mickey Mantle for the second consecutive year and won. Mantle had led almost all the way in 1956 only to have Williams to take the lead September 14-21.

1956 BA race: Mantle v. Williams. Tuesday, November 22, 2016

Their game 1957 logs:
Mantle
Williams

This link has that data combined, which is how the graph was created. The first two date columns have the dates when they played. Column three has all dates during the regular season. For dates when they played doubleheaders, the BA after game two is shown. For dates when they did not play, the BA of the most recent game is carried forward.

Williams had the highest BA six times: 1941, 1942, 1947, 1948, 1957, 1958. Mantle led once, 1956.