UPDATE June 8, 2020: The final season of Sword Art Online: Alicization ~War of Underworld~ has been confirmed to be coming out on July 11, 2020. The remainder of this article remains as initially published.
Sword Art Online Season 3 has been confirmed to be taking a TV broadcasting break. The good news is that the Sword Art Online: Alicization anime is very likely to return in the second half of 2019.
The news of the Sword Art Online: Alicization break came indirectly via the website for the Monogatari Series anime. It was announced that Zoku Owarimonogatari & Monogatari Series Selection will begin airing episodes on multiple Japanese TV channels starting April 6 at midnight.
The reason this is related to Sword Art Online: Alicization is because SAO Season 3 currently airs during those time slots on the TV schedule.
The Sword Art Online: Alicization hiatus is not a surprise. Before the third season began airing, A-1 Pictures had already confirmed that the entire story for Alicization would be composed of four cours (a cour is a three-month unit of broadcasting time based on the physical seasons).
What’s more, Matsuoka Yoshitsugu (voice actor for Kirito) had previously mentioned that Alicization’s four cours would be “split into 2 x 2-cour parts.”
Sword Art Online Season 3 will officially go on hiatus starting in April 2019. Therefore, the SAO: Alicization anime is likely to return in July 2019 or October 2019. (This information has not been confirmed yet and is simply an educated guess based on knowledge of the anime industry).
Despite taking such a long break, the second part of the Alicization anime is unlikely to be labeled as Sword Art Online Season 4. Instead, that right will be reserved for the Sword Art Online: Unital Ring anime, which could adapt the final story arc of author Reki Kawahara’s SAO light novel series.
Warning: Reading the plot summary for Unital Ring necessarily contains some indirect spoilers for Alicization.
A two-cour break for Alicization would be similar to Attack On Titan Season 3, which is airing a second half in April 2019 rather than being called Attack On Titan Season 4 (which, similar to SAO, will likely be reserved for adapting the ending of the Shingeki no Kyojin manga series).
This is called a split-cour anime, which is when a single anime season takes a long broadcasting break but is still considered a single season.