it was her husband who rushed her to near hospital that afte

你好.应该是：

it was her husband that rushed her to a nearby hospital that afternoon..
或者是：
it was her husband that rushed her to the nearest hospital that afternoon..
如果用around,句子就是：
it was her husband that rushed her to a hospital around that afternoon..(around是副词,只能做后置定语)

语感告诉我,强调句结构的连接词最好是that.

感觉应该改成to the nearest hospital,您的句子中near的用法我感觉不对

你的翻译应该是没问题的，最好还是用near吧，around有点怪怪的。

near 作为形容词时很少这么用，通常是 the nearer 或者 the nearest。意思是更近的或者最近的。
这个句子若是要直译“附近的”，用a nearby hospital更为合适，或者复杂点用a surrounded hospital也没错。
around是介词，不可以用在这个位置。