Keeping Salvadoran Families Together Act This bill amends the Immigration and Nationality Act to permit an alien who is a national of El Salvador in temporary protected status (TPS) to apply for legal permanent resident status if such alien: is eligible for permanent resident status, applies for adjustment within three years, was granted or was eligible for TPS status, and has been continuously physically present in the United States for at least three years. (TPS designations permit eligible nationals of designated counties affected by armed conflict or natural disasters to temporarily reside and work in the United States.) The bill: waives certain grounds of inadmissibility; authorizes the waiver of the continuous physical presence requirement if an alien's removal would cause extreme hardship to the alien or to the alien's spouse, children, parents, or domestic partner; authorizes an alien who has applied for status adjustment to work; and authorizes an alien who has been ordered removed or granted voluntary departure to apply for status adjustment. An alien's spouse, parent, or unmarried child shall have his or her status adjusted to legal permanent resident if such person is eligible for status adjustment and applies within three years.