Using Intimidate to demoralize an opponent can only get them shaken, by default. You cannot use it repeatedly to achieve greater levels of fear.
First, in general effects do not self-stack, that is, repeated applications of the same effect have no further effect but just extend the duration. This is explicitly called out in the case of magic effects; for non-magic effects, the rules are more vague, but Intimidate has those retry rules.
The Intimidate retry rules read as saying that most NPCs will not respond to a second Intimidate check—either the first was successful and they are as frightened as they are going to be via the skill, or it didn’t and they are now resolute. Which seems kind of dumb to me but then, the social skills are really, truly dumb in many ways more serious than this.
In addition to being my reading of the rules, this “one shot” interpretation definitely seems to be there consensus online, in my experience. I have never played with or spoken with anyone who read the rules differently (houseruled them differently, yes, but as an explicit houserule).
But your bard need not necessarily give up here. There are feats and effects that can improve the demoralize effect to frighten or panic foes, even force them to cower in fear. The fearsome magic armor property and the Imperious Command feat from Drow of the Underdark, as well as the Never Outnumbered skill trick from Complete Scoundrel, are great places to start with that kind of thing. Also, the Doomspeak feat from Champions of Ruin is a more magical form of debuff, but it’s thematically similar, bard-specific, and absolutely devastating, so worth a look. For more, I recommend the Intimidate section of the Fear Handbook.