Is the following sentence correct?

Rob is not at school today, but said he would come tomorrow.

Notice that the version above version does not have a subject before said. Should it be:

Rob is not at school today, but he said he would come tomorrow.

The first option sounds quite right to me, but I would like to know if there is any rule to decide whether or not we can omit the subject of a sentence - especially when the tenses in the sentences are not the same? I'm most interested in what the rule is, not just whether we need to put the subject in that sentence or not.