Good news, there's more Livy! As well as 1-10, books 21-45 have survived pretty much intact, taking you up to around 166 BC. You can read them for free here. For the gap in the middle, or at least bits of it (including the First Punic War), you should turn to Polybius, whose histories are just are readable as Livy, although perhaps a bit less sensational.

After that there's another gap until you get to Sallust's narratives of the wars with Jugurtha and Catiline, both of which are quite enjoyable, I think.

If you don't mind stepping away from history into biography then I'd totally recommend Plutarch's Parallel Lives. He covers lots of the famous figures of the republic, especially clustered around the dramatic events of the first century BC. The standards for what we would call 'historical accuracy' were a bit more relaxed for biography - biographers were more interested in exploring the personalities and characters of the individuals they were covering, and they weren't above exaggerating some of the exciting bits for dramatic effect. Pick people who interest you then dive in!

Hope that helps. I'd be happy to answer further questions if you still have any.