Supergirl Archives, Volumes One and Two covers Action Comics #252-285 (1959-1961)
The Silver Age Supergirl covers Action Comics #286-376 (1962-1969)
Both the archives and the omnibuses focus on Supergirl's solo feature in Action Comics, but Supergirl made guest appearances in other DC comics across the "Superman" family of titles between 1959 and 1961. The "DC Finest" edition roughly equates to the Supergirl solo stories published in the two archive editions (through #288) plus the guest appearances.
Superman's Pal, Jimmy Olsen #40 (October 1959)
Superman's Girlfriend, Lois Lane #14 (January 1960)
Action Comics #260* (January 1960)
Superboy #80 (April 1960)
Superman's Pal, Jimmy Olsen #46 (July 1960)
Superman #139 (August 1960)
Superman #140 (October 1960)
Superman's Girlfriend, Lois Lane #20 (October 1960)
Action Comics #270* (November 1960)
Adventure Comics #278 (November 1960)
Action Comics #273* (February 1961)
Superman's Pal, Jimmy Olsen #51 (March 1961)
Superman #144 (April 1961)
Action Comics #276* (May 1961)
Action Comics #281* (October 1961)
Superman's Pal, Jimmy Olsen #57 (December 1961)
Replies
SUPERMAN'S PAL, JIMMY OLSEN #40 - "Jimmy Olsen, Supergirl's Pal!"
"As readers of Action Comics know, Supergirl has recently come to Earth posessing all the super powers of her cousin Superman! For certain reasons, the very existance of the Girl of Steel is being kept secret from the world! But one day, Supergirl decides to reveal herself to Superman's trusted pal, Jimmy Olsen! What happens then will utterly amaze you as the cub reporter becomes... Jimmy Olsen, Supergirl's Pal!"
This is Supergirl's first guest appearance in another title, and her sixth story ever. In it, Jimmy runs afoul of a con man running a fake sideshow. Jimmy threatens to expose him and the con man threatens Jimmy's life. Then, Jimmy accidently blinds himself (as one does). Superman is unavailable, but he has arranged for Supergirl to cover for him, which includes responding to Jimmy's emergency watch signal. (Zee! Zee! Zee!) Supergirl catches up to him just as the con man is shoving Jimmy off a bridge. She decides, because Jimmy is superman's pal, that it would be okay to reveal her identity to him, so she swoops in to save him. Once safely on the ground, she relates her orgin story to him but, skeptical, he doesn't believe her. He thinks that she is a confederate of the con man playing a hoax on him. Supergirl is worried that, if he doesn't believe she's really who she says she is, that he won't activate his signal watch the next time the con man tries to kill him. So she sets about trying to prove to the temporarily blinded Jimmy that she's Supergirl. She performs seven different super-feats, but Jimmy is always able to think of a way she might have faked it. Just then, Superman returns from his mission and Supergirl uses a puff of her super breath to activate the emergency watch, drawing superman to the spot. Superman apprehends the con man (who is also a killer, BTW), and Jimmy relates the story of how he tried to trick him into believing there is a "Supergirl." (We don't find out what Superman has to say to Kara about this.)