Welp, if other options were considered, then they weren't considered for long! If you follow the narrative laid out by Suetonius in The Lives of the Twelve Caesars, Caligula was assassinated on January 24th, and Claudius was reluctantly appointed his successor on... January 24th.
Allegedly he actually his from the soldiers scouring the royal households (they were actively looking for a successor, more on that later), and he was found hiding behind a curtain. Supposedly, (and true to Suetonius' delightfully salacious style) he was found because he perfume was so strong that the soldiers smelled him before they saw him. He was promptly told that he was the new emperor, and that was that.
When thinking about this, you must consider that the murder of an emperor at this stage in the empire was a new thing, and it really created absolute chaos. Claudius was murdered by a former Praetorian Guard, the elite military of Rome, after games at the Coliseum, surround by other guards, children and dancers. I've heard it compared to the US president being murdered by a spurned member of the CIA, while escorted by the Secret Service, just after leaving the Super Bowl, surrounded by the cast of Glee (it's becoming a little bit of a dated metaphor).
Suetonius also related that the cohors Batavorum, the German bodyguards that personally guarded the emperor went, to use a slightly anachronistic phrase, berserk upon hearing of Caligula's death; to them, originating from a strict tribal structure in Iron Age Germany, the death of their "chief" was an absolute worse case scenario. They actually killed several unlucky senators as they raged about looking for bloody justice.
Furthermore, you must consider why the quick turn around of emperor to emperor - the army. The military, primarily the Praetorian Guard, became, frankly, a cancer to the Imperial rule, in that their control of it grew as time went on. Augustus had seized an empire on the loyalty of his troops, but monetary gain backed up that loyalty. The Romans in charge knew this; Septimius Severus, another usurper and last-man-standing at the end of the Year of the Five Emperors in 193CE, upon his death bed told his two sons, Geta and Caracalla "get along with each other, and pay the army" (Carcalla pretty quickly forgot the first piece of advice and had his brother murdered). Without the support of the Praetorian Guard and army (usually backed up by a significant pay raise), it became that an emperor wouldn't last long. It got so bad that eventually the Praetorian Guard actually auctioned off the title of Emperor (to Didius Julianus, who then was usurped by the aforementioned Septimius Severus). But the seeds of this kind of greed were sown when the empire was born, and relating back to the initial question - if there was no emperor on the throne, then no-one in the army was getting paid. That's what it boiled down to; the soldiers needed paying, they were paid by the emperor as head of state, therefore there needed to be an emperor, and quickly. I'm sure that, as the news spread quickly, some briefly considered a glorious return to Auld lang Syne, but with less than a day before there was a defacto Julio-Claudian emperor on the throne, this would be have been little more than a fleeting pipe dream.