Yes, the toad can bite the target they're already holding.
The attacks listed in a monster statblock represent separate options that it has.
If the Swallow attack succeeds, the toad must swallow the victim. It says, very plainly, "if the attack hits, the target is swallowed."
However, it still has the Bite attack, which has a 5' reach (which the victim is definitely inside) and this restriction:
Until this grapple ends, the target is restrained, and the toad can't bite another target.
This implies that it can bite the target it's already holding (and in fact it has to in order to swallow them, since that uses a bite attack). This attack would have advantage since the target is restrained.
Note that even if one ruled the other way (that using Bite against an already-grappled target would automatically be an attempt to Swallow), the toad could still just release them and immediately Bite again. However, in that case the creature wouldn't be restrained.
Note also that if the toad bites a creature too big for it to swallow, it does still get to grapple and restrain it, and could bite it again on subsequent turns.