"Joe Pickett" abandoning Spectrum Originals after its first season is ultimately unsurprising. While Spectrum Originals still exists, and even aired new episodes of a few of its original series as recently as June of 2022, shows jumping ship from the service is not a new phenomenon. For example, before Season 3 of "L.A.'s Finest" was canceled, the show began airing on Fox following two seasons of Spectrum Originals exclusivity.
Its somewhat complex distribution history seems to have impacted precisely where "Joe Pickett" is available digitally today. For example, on Vudu, only Season 1 is available for purchase. Meanwhile, the show is completely absent from other major storefronts like Google Play. The only platform on which those entirely uninterested in a Paramount+ subscription can still obtain episodes of both seasons of "Joe Pickett" is Apple's iTunes store.
The only way to watch both "Joe Pickett" Seasons 1 and 2, then, is either with a Paramount+ subscription or through the iTunes store where individual episodes and seasons can be purchased à la carte.