その後は特に気にかけることもなく時間がすぎたということです。
I understand it as "After that, without discovering anything new we spent time on it" but I'm really not sure.
context of the sentence:
"足元に黒く光る変わった石が
今回、発見された隕石は６年前の秋、三津村さんが自宅近くの畑で野菜の収穫作業をしていた際に偶然見つけました。 休憩のため腰を下ろそうとした時、足元に黒く光る変わった石があるのに気づき、持ち上げたところ重かったため、「珍しい」と思って持ち帰り、玄関に飾ったということです。 訪ねてきた知人に「隕石じゃないか」と指摘され、磁石を近づけてくっつくことも確認しましたが、その後は特に気にかけることもなく時間がすぎたということです。"
「その後{ご}は特{とく}に気{き}にかけることもなく時間{じかん}がすぎたということです。」
You have translated that sentence to:
"After that, without discovering anything new we spent time on it."
I wonder where you get "discovering anything new" from. There is no word/phrase even close it in the original sentence.
There is no equivalent of "spent time on it" in the original, either, but this is far better than the "discovering" part.
「気にかける」 means "to be concerned about", "to pay (extra) attention to", etc.
「時間がすぎた」 simply means "time passed by"
Thus, my own TL would be:
"(He says that), after that, he was not particularly concerned about it and time just passed by."
"without discovering anything new" would be literally translated as "なんの新しい発見もなしに".
気にかける means "care about", "be concerned about" and so on. https://eow.alc.co.jp/search?q=%e6%b0%97%e3%81%ab%e6%8e%9b%e3%81%91%e3%82%8b&ref=wl
I think "without caring about the stone" or "without paying attention to the stone" are appropriate. And "he spent time" is correct.
