Complex sentence: Joplin took a more active role in putting together the Full Tilt Boogie Band than she did with the Kozmic Blues Band, and she believed that hiring the remainder of Till's band would provide her with the powerhouse boogie outfit she needed to back her.
Split into simpler sentences: Joplin took a more active role in putting together the Full Tilt Boogie Band than she did with her prior group, the Kozmic Blues Band. Joplin was convinced that hiring the remainder of Till's band would provide her with the powerhouse boogie outfit she needed to back her.