I think a feat is too powerful. It's not related to multi-classing really either. It's the gap that's the issue, not the missing attack. If you focus on the missing attack, you've shifted to "let's reward multiclassing these classes". Putting something there makes sense, but not tying that something to the power level of an extra attack. Which is why I proposed getting what you missed out on for multi-classing when you first multi-classed, so it's tied to multiclassing itself, and not the extra attack.
So for example, if your second class is fighter under RAW you get:
Armor: Light Armor, Medium Armor, shields
Weapons: Simple Weapons, Martial Weapons
Tools: none
Skills: none
But the Fighter at first level gets these proficiency:
Armor: Light Armor, Medium Armor, Heavy Armor, shields
Weapons: Simple Weapons, Martial Weapons
Tools: none
Skills: Choose two skills from Acrobatics, Animal Handling, Athletics, History, Insight, Intimidation, Perception, and Survival
So at the "empty" multiclassing level you'd get Heavy Armor proficiency and two skills from that list.
You could also add back in the missing saving throw proficiencies. Normally a first level fighter gets Strength, Constitution saving throw proficiency so you could add those in at that level as well, if you feel the armor and skills is not enough (though I think adding BOTH saves is probably excessive if your first class had no overlaps with those).