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I think you'll need a phono stage and a headphone amp.
I can't think of a headphone amp with a phono stage in it, but then again, they may well exist. Otherwise, your headphone amp will be expecting a line level input, not the minuscule signal from a MC cart.
Edit:
Cambridge Audio duo do the business?
oh.
forget about all that, you have already realised this and found one that does exist - looks fine!
So. I did the thing. I have purchased a 2nd hand Inspire/quest/rega/Ortofon yaddah,yaddah somesuch, because I saw it, and it gave me happy vibes, like the typewriter in Garth Marenghi's TerrorTome it drew me in. More benignity, than horror obviously. Sexy wood veneer benignity. Torlyte benevolence.
But also cos I decided this was going to be a learning experience, I'd give myself some headaches along the way.
Tl:dr. Headphone amp for low output MC cartridge (methinks?), namely Ortofon Quintet Red? Cambridge Audio Duo do the business?
Oh yeah, headphone advice heeded.