Since breaking down the 162-game season into six 27-game sections is a better measure than just saying, "The Sox won their most games ever in April", here are the best "First 27 W-L" since 1961, and the final record, when they went to 162 games:
2005- 20-7, 99-63
1965- 19-8, 95-67
1967- 19-8, 89-73
2006- 19-8, 90-72
1964- 18-9, 98-64
1973- 18-9, 77-85
1982- 18-9, 87-75
1963- 17-10, 94-68
1972- 17-10, 87-67 (strike shortened)
1977- 17-10, 90-72
1990- 17-10, 94-68
1992- 17-10, 86-76
2000- 17-10, 95-67
2002- 17-10, 81-81
2004- 17-10, 83-79
The 2016 team has two more games, and will win 17, 18, or 19 games. Most of these years, the Sox end up having very good win totals. Does this mean much in the era of the unbalanced schedule and heavy parity in the AL? Maybe not. But the Sox have probably already played themselves into season-long contention just by starting 16-6.