Is there another term for "in fact", "actually"?

Question

I feel like I am blanking on a very common conjunction word. よく聞く言葉みたいな気がしますが、どうしても思い出せないんです。I am looking for a word that means "in fact", "actually", as in:

You can call me by my first name. Actually, please call me by my first name.
ファーストネームで呼んでいいぞ。______　ファーストネームで呼んでくれえ。

I don't think it is a good idea to go into the school building at night when nobody is there. In fact, it is illegal if you do and I am going to call the police.
夜に誰もいない校舎に入るのは良くないと思うよ。______　入ったら違法行為で、通報する。

Dictionaries and Google Translate have been of no help at all, because these English phrases have multiple meanings, and the dictionaries all give me words for one sense of these two phrases: 「実際に」, but these phrases mean something totally different in the contexts given above, more like "on the contrary".

I can only think of 「それどころか」. But I thought there was a more colloquial, informal phrase/word for it. I seem to vaguely recall it was something shorter, crispier. Is there such a word? For some weird reason I keep thinking it sounds like 「もはや」...

Answer 1

ああ、思い出しました。もしかして「むしろ」かな。

ファーストネームで呼んでいいぞ。むしろファーストネームで呼んでくれえ。

夜に誰もいない校舎に入るのは良くないと思うよ。むしろ入ったら違法行為で、通報する。