For Phoenicians in America, no, there is no evidence.

For West Africa, there is. We have preserved (via Greece sources) an account of the Voyage of Hanno link and really have no reason to question the possibility. The Phoenicians certainly possessed the technology by the 6th century BCE, and were sailing to the Straits of Gibraltar and up the Atlantic coast of the Iberian peninsula even in the 8th century BCE. Cadiz is a Phoenician foundation dating to this period, along with a string of smaller settlements along the Costa del Sol of Andalusia. Carthage, too, was founded in the 8th century, about 750 BCE, followed shortly after by Motya (western Sicily) and sites on Sardinia.

We have some limited archaeological evidence indicating Phoenician presence (perhaps even settlement) along the Atlantic coast of Western Africa. Along the Bou Regreg estuary, in modern Rabat, Morocco, a good deal of Phoenician red slip pottery has been found. This site is called Sala and is thought to have been an emporion, or trading enclave. The pottery is associated with a large building of ashlar blocks which has a form similar to that found at other Phoenician sites and often called a "storeroom." This material dates to the 7th and 6th centuries BCE.

Almost 400km further south, more Phoenician material has been found on the island of Mogador, in the bay of Essauoira. A large amount of Phoenician red slip pottery has been found here as well, with multiples examples featuring Phoenician inscriptions. Greek and Cypriot pottery also appears, almost certainly carried by Phoenician (and not Greek) vessels. Both of these sites are usually thought to be seasonal depots for facilitating trade or the exchange of prestige objects with the west African interior.

For more on this topic, see Glenn Markoe's excellent book on the Phoenicians, titled simply Phoenicians (2000). He also discusses the feasibility of Phoenician circumnavigation of Africa and voyages to Atlantic destinations beyond the Straits.