How can I use one casting of vampiric touch more often?
As far as I understand it, vampiric touch does the following:
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Casting the spell takes an action (leaving the caster a bonus action, a reaction and a move left to do in no particular order)
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For the duration of said minute (10 rounds) one has a chance to apply 3d6 necrotic damage and heal 1/2 of that, up to ten times via "melee spell attack".
How could I acquire more "melee spell attacks" per round?
Rangers', Fighters', Paladins' and Bards' (College of Valor) Extra Attack doesn't qualify and neither do Monks' Flurry of Blows, because none are "melee spell attacks". Even the spell haste won't work and that is practically a minute-long use of Action Surge.
Best Answer
On the first turn, Quicken Spell will let you attack twice with it. By casting it as a bonus action (which includes attacking with it), you can free up your action, allowing you to attack with it again.
Other than that, the Fighter's Action Surge ability is your only option here.
While there are various methods of speeding up the casting of a spell, Vampiric Touch gives you an extra use for your action. As such, only abilities that let you make multiple actions, not multiple attacks, can help you use it more than once. The fact that it involves a melee spell attack is unimportant.
As you've said, Haste won't work, since it specifies what actions can be used with its extra action. Time Stop won't work either (well, it will let you use Vampiric Touch on yourself, if that was a thing you wanted to do). Action Surge is the only ability that will give you an extra action with no restrictions.
Note that even with Action Surge, using it more than once in a turn is still potentially shaky. It says you can make the attack again on "each of your turns", which could be taken to imply that you can only use it once on each turn.