A full attack is not something granted because you have multiple attacks, it is the required action to take advantage of them. You can always make a full-attack, even if you see no benefit for doing so.
A character with haste, by definition, has multiple attacks. Thus, they not only can use a full attack (as usual), but they must if they wish to use the extra attack granted by haste.
Thorough rules breakdown:
SRD > Combat > Actions in Combat
The Combat Round
In a normal round, you can [...] perform a full-round action.
This establishes that full-round actions are not any kind of special case; they are a normal option for how you spend your turn.
A full-round action requires an entire round to complete. Thus, it can’t be coupled with a standard or a move action, though if it does not involve moving any distance, you can take a 5-foot step.
This establishes some parameters for the use of full-round actions. Note that it does not put any restrictions on the use of full-round actions other than the things you are not allowed to use at the same time. That is, a full-round action is always an option, provided you have not used a move or standard action this round (and, in the case of full-round actions that involve movement, have not used a 5-ft-step).
If you get more than one attack per round because your base attack bonus is high enough, because you fight with two weapons or a double weapon or for some special reason you must use a full-round action to get your additional attacks.
This establishes situations in which you are required to use a full attack. It does not discuss any situation in which you are required not to use a full attack. Notably, it never puts any special limitations on when you use a full attack beyond the usual requirements full-round actions.
The remainder of the full attack entry1 is irrelevant here. It just discusses various details related to the actual use of a full attack once you have chosen to use one.
And that’s literally every relevant rule on this topic. None of them put any requirement on the use of a full attack, aside from the normal requirements for full-round actions (i.e. not in the same turn as a move or standard action). Therefore, since full-round actions are a normal option for your turn, and a full attack is a full-round action like any other, and nothing puts any limitations on it,
You may always use a full attack, regardless of number of attacks
Yes, you can use a full attack even when you only get one attack. Doing so prevents you from using your move action, and provides absolutely no benefit, but you’re still allowed to, if you want. For instance, if you had a feat or class feature that started “When you take a full attack action,” that feat or class feature would be triggered by such a single-attack full attack, but not by a standard-action attack (even though they’re the same number of attacks).
Also, for completeness, this is all moot in the case of haste – if you have haste, you have more than one attack anyway, and therefore are required to use a full attack in order to get that extra attack. The rules don’t care where the attacks are coming from, and in fact specifically talk about multiple attacks “because your base attack bonus is high enough, because you fight with two weapons or a double weapon, or for some special reason,” providing an explicit catch-all for things like haste.
1 For reference, the rest of the full attack entry:
The only movement you can take during a full attack is a 5-foot step. You may take the step before, after, or between your attacks.
If you get multiple attacks because your base attack bonus is high enough, you must make the attacks in order from highest bonus to lowest. If you are using two weapons, you can strike with either weapon first. If you are using a double weapon, you can strike with either part of the weapon first.