How did they find out about the cab ride? I don't think Oswald told them. Did Whaley see Oswald on TV and call the authorities?
Whaley said he saw the picture of Oswald in the paper the next day and told his boss/superior at the cab company that he had given Oswald a ride. He said his superior called the police and they came down and got him.
From his testimony:
Mr. BALL. Later that day [of the assassination] did you--were you called down to the police department?
Mr. WHALEY. No, sir.
Mr. BALL. Were you the next day?
Mr. WHALEY. No, sir; they came and got me, sir, the next day after I told my superior when I saw in the paper his picture, I told my superiors that that had been my passenger that day at noon. They called up the police and they came up and got me.
Mr. BALL. When you saw in the newspaper the picture of the man?
Mr. WHALEY. Yes, sir.
Mr. BALL. You went to your superior and told him you thought he was your passenger?
Mr. WHALEY. Yes, sir.
Mr. BALL. Did the Dallas police come out to see you?
Mr. WHALEY. Yes, sir.
Mr. BALL. Or FBI agents?
Mr. WHALEY. The Dallas police came down and took me down and the FBI was waiting there.
Source:
https://www.jfk-assassination.net/russ/testimony/whaley1.htmFritz said he asked Oswald about getting a cab to go home. It's not clear who told him about the ride though.
Mr. FRITZ. He [i.e., Oswald] told he that was the transfer the busdriver had given him when he caught the bus to go home. But he had told me if you will remember in our previous conversation that he rode the bus or on North Beckley and had walked home but in the meantime, sometime had told me about him riding a cab.
So, when I asked him about a cab ride if he had ridden in a cab he said yes, he had, he told me wrong about the bus, he had rode a cab. He said the reason he changed, that he rode the bus for a short distance, and the crowd was so heavy and traffic was so bad that he got out and caught a cab, and I asked him some other questions about the cab and I asked him what happened there when he caught the cab and he said there was a lady trying to catch a cab and he told the busdriver, the busdriver told him to tell the lady to catch the cab behind him and he said he rode that cab over near his home, he rode home in a cab.
Source:
https://www.jfk-assassination.net/russ/testimony/fritz1.htm