Vampiric Touch says,
The touch of your shadow-wreathed hand can siphon force from others to heal your wounds. Make a melee spell attack against a creature within your reach. On a hit, the target takes 3d6 necrotic damage, and you regain hit points equal to half the amount of necrotic damage dealt. Until the spell ends, you can make the attack again on each of your turns as an action.
Round 1 you cast vampiric touch and make a melee spell attack for 3d6 necrotic damage.
Round 2 while still concentrating on vampiric touch you cast Inflict Wounds as a 3rd level spell for 5d10 necrotic damage.
Being that Vampiric Touch is still functioning, you are making a melee spell attack, and it is dealing necrotic damage... Does the necrotic damage from Inflict Wounds siphon health?
(Obviously the 3d6 and 5d10 wouldn't stack because of action economy, etc.)
I am sure that a player could find a better use of two 3rd level spell slots, but I have taken up a kind of awkward place as an arcane healer/tank for the group and every bit of preservation helps.