Kellogg signed with Chicago and was assigned to the Low-A Eugene Emeralds, where he spent the whole season, posting an 0–1 record with a 4.98 ERA in 21.1 innings pitched.

In 2017, Kellogg spent the season with the High-A Myrtle Beach Pelicans where he struggled, going 5–7 with a 5.12 ERA in 23 games (twenty starts).

[2] In 2018, Kellogg returned to Myrtle Beach, where he moved to the bullpen, compiling a 3–1 record with a 3.58 ERA in 37 relief appearances.

[3] In 2019, Kellogg remained in Myrtle Beach for a third straight season, registering a 2-8 record and 4.54 ERA with 48 strikeouts in 23 appearances.

[6] In 2021, Kellogg played in 20 games between the Double-A Tennessee Smokies and the Triple-A Iowa Cubs, recording a 4.62 ERA with 42 strikeouts in 64 innings of work.