He went to the University of Kentucky for college. As a junior in 1994, he hit .445 with 23 homers and led the Southeastern Conference in batting average. He had the 4th-highest average in NCAA Division I, was 4th with 203 total bases, tied for 9th with 24 doubles, was 4th with 23 homers. He was drafted by the White Sox twice, first in the 32nd round in the 1993 amateur draft, and then in the 4th round in the 1994 amateur draft.
He never hit under .320 in the minors before coming up to the majors. At the Rookie Level, at Sarasota in the GCL, he only needed four games, posting a .467/.579/.733 line, to convince them he should be moved up. He also clearly outclassed the league at Hickory in the class A SAL, where he put up a line of .393/.481/.598.
The next year, 1995, found him at Prince William in the Carolina League, where he posted a .348/~.410/.455 line. Moved up to Birmingham in the AA Southern League midway in the season, he posted a .320/~.383/.431 there.
In 1996, he spent the season at AAA Nashville in the American Association, hitting .325/~.375/.486. He was second to Dmitri Young in the league's batting race, made the AA All-Star team, was picked as the league's #3 prospect by Baseball America and was also named the top batting prospect in the league. In spite of those numbers, he was slated to stay there another year. In 1997, at age 25, he put up almost exactly the same numbers, .327/~.385/.486, and finally earned a shot at the bigs. He was error-free in the outfield, led the AA with 88 runs and again was second in average, two points behind teammate Magglio Ordonez. He was again named to the league's All-Star team.
He appeared in 19 games with the 1997 White Sox, hitting .263/.263/.368 with 1 double and 1 home run. The White Sox that year went 80-81, down from the previous year, but still second in the division. Abbott's performance was good enough that he stayed on the team in 1998.
In 1998, he hit .279/.298/.492 with 12 home runs in 244 at-bats. He was the fourth outfielder on the team, playing in center, right, and left fields. The White Sox went 80-82, almost the same as the previous year.
Starting slowly in 1999, with a .158 average in 17 games, he found himself back at AAA Charlotte in the International League. He showed that he was still too good for minor league pitching, putting up numbers of .318/~.359/.509. He and Charlotte went to the Triple A World Series.
2000 was a good year for Abbott, as he was a contributor to the division winners. He put up a .274/.343/.395 line in 80 games, as he again served as the fourth outfielder. The team won 95 games. He had one post-season at-bat.
After the season, he was traded to the Florida Marlins. He was 28, with a couple of seasons under his belt as a fourth outfielder on a decent team, and it might have been expected that he would be a regular outfielder for several years with the Marlins. It was not to be.
Jeff started the season with knee problems, and was to bounce between various teams the whole season. He showed he still dominated AAA ball by posting .320/~.347/.526 at Calgary in the PCL, and had seven dominant games in AA at Portland of the Eastern League with .462/~.501/.731. He was also in single A Brevard County of the Florida State League for 3 games, hitting .250. Up in the majors, he appeared in 28 games, with results of .262/.326/.333. While the minor league performances were good, the Marlins let him go after the season.
The Boston Red Sox gave him a shot at AAA Pawtucket of the International League, and for once he was under .300: .283/.331/.447. He was there the entire season of 2002.
In 2003, he appeared in only 4 games at AAA Tacoma of the PCL, getting 2 hits in 15 at-bats, and he was done.