Spain’s socialist party has begun weighing up its options for government after storming past its traditional conservative rival in Sunday’s snap general election but failing to win a majority.

The prime minister, Pedro Sánchez, led his socialist PSOE party to its first victory since 2008, winning 123 seats and taking 29% of the vote.

The rightwing People’s party (PP) endured a catastrophic night as its seat count plummeted from 137 to just 66, while the poll also marked a breakthrough for the far-right Vox party, which took 24 seats to enter the national parliament for the first time.

The PSOE also routed the PP in the upper house of the Spanish parliament, where it took control by winning 121 senators – up from 43 three years ago. The PP’s headcount was slashed from 130 to 56.

With Spain’s third election in four years producing another hung parliament, speculation is mounting as to how the PSOE will secure the 176 seats needed to form a government in the 350-seat congress of deputies.

At the moment, the socialists are aiming to govern alone, albeit with the continuing support of the anti-austerity Unidas Podemos and its allies.

“We think we’ve got more than enough support to keep steering this ship,” the deputy prime minister, Carmen Calvo, told Cadena Ser radio on Monday morning.

“We’re very aware that Unidas Podemos has helped us a lot and backs our progressive aims. But we think this is the course we need to follow.”

Even with the support of Podemos and its allies, Sánchez would have only 165 seats and would need to rely on smaller regional and nationalist parties.

He will be keen to avoid turning again to the two Catalan separatist parties who backed him in the successful no-confidence motion that removed the PP from office in June last year and allowed the PSOE to take power.

The two parties – the Catalan Republican Left and Together for Catalonia – triggered the snap election by siding with the Spanish right to reject his 2019 budget.

Such a move would also allow Sánchez’s opponents to deploy their familiar argument that the prime minister is too beholden to Catalan separatists and ought to take a less conciliatory approach to the independence issue.