Connecting Students to Great Jobs Act This bill directs the Department of Labor and the Department of Education (ED) to award career pathway grants and provide assistance for career pathway programs. Labor must award competitive grants to eligible institutions of higher education and vocational institutions to develop, offer, improve, and provide educational or career pathway programs for workers. Grant funds must be used for specified purposes, including for career pathway programs that provide relevant job training to meet the needs of employers in high-skill, high-wage, or in demand sectors or occupations and that may include registered apprenticeship programs. The bill requires ED to carry out a program to award START federal Pell Grants to students in short-term programs that prepare participants for employment opportunities in high-skill, high-wage, or in demand sectors or occupations in a geographic area in which the programs are offered. The bill also directs ED to (1) award grants to states to establish statewide career pathways coalitions, (2) establish a career pathways innovation grant program to support career pathways in high school and career exploration in the middle grades, and (3) provide technical assistance.