On 9 October, she again performed at Bobino, this time for three weeks as the headline performer, where she promoted her latest releases "Du moment qu'on s'aime", that peaked the charts same week, and "Come prima". Topping charts in January 1959, where it remained during most of winter, "Come prima" proved to be an ultimate holiday hit of France and Belgium, as Dalida could be seen promoting it on a Christmas-themed television set. Record's B side song "Si je pouvais revivre un jour ma vie" also achieved success topping Canadian charts. According to her biographer Catherine Rihoit, her live television appearances "got her into the palms of society, keeping her position as a favorite singer of all age groups in the francophonie", and marked an era of late '50s television shows. On 26 December 1958 Dalida was in New York with Morisse where they met Norman Granz, the American impresario of Ella Fitzgerald, who invited her to Hollywood and offered a fifteen-year contract to launch her career in the USA. She quickly rejected the offer, saying that she wanted to focus on her musical career in France where she was already well-known with a secure fan base.
Dalida toured extensively in 1959, playing sold-out dates in France, Egypt, Italy, and Germany. As her fame spread outside France, she started to record songs in other languages to cater for these new audiences. In February, during a TV appearance, she performed her recent version of "Hava Nagila". On 2 March, Minister of informations awarded her, alongside Yves Montand, with the Bravos du music hall, the most prestigious music award in France, for the most popular singer in France. During the summer, she covered her own recordings "Ne joue pas" and "C'est ça l'amore" in Flemish as "Speel niet met m'n hart" and "Ik zing amore", respectively, the only time in her career she sang in Flemish.
In late summer, Dalida was back in the studio to record her first major international hit. In the period from 1958 to 1959, "Le jour ou la pluie viendra" was recorded in three languages, which led Dalida straight to the top three in six different European countries. The German language version "Am Tag als der Regen kam" topped German charts for ten weeks in September and October, earning her another gold disc. It was the best-selling record of the year in Germany, and remained one of the most successful songs in history of the country. During the closing night of the Berlin Film Festival on 28 September 1959, she was presented with a Goldener Löwe award by RTL, for the best-selling musical artist of the year in Germany, and saluted with fanfare playing "Am Tag als der Regen kam" verses. The song was her first international recognition, reaching a half-million accumulated copies and eventually remaining one of her biggest success in Germany.
In the course of 1959, Dalida collected five Top 10 hits in French charts, most notably "Ciao, ciao Bambina" and "Guitare et tambourin"; both earning gold discs. In Italy, RAI awarded her with Oscar di popolarità and Lupo d'oro awards for the best-selling musical artist of the year in the country. Those were her first two foreign awards that furthered her international recognition. Dalida also performed in successful sold out concerts in Berlin, Athens and Cairo, delivering a sentimental performance in front of a crowd in Rivoli cinema that she frequented as a child. On 23 September 1959, Dalida sang in a successful three-week run at Parisian Théâtre de l'Étoile, where a jukebox was installed in recognition of her being named Mademoiselle Jukebox, the most listened to artist on jukeboxes in France. By the end of the year, she released her fifth and sixth albums; Le disque d'or de Dalida and Love in Portofino, and had already sold three and a half million records, highest among all European artists.