You don't even need to be connecting to an Amtrak train to gain access to the CA's or the Metro Lounges. As long as you have a same day First Class ticket or a sleeper ticket or stubs for either, you are entitled to use the lounge. And by same day, that means that you were traveling that day in either a sleeper or FC.
So for example, if you had booked a sleeper for yesterday's Capitol Limited out of Chicago, when you arrive into DC you are holding a "same day" ticket stub for a sleeper. So you could then use the CA in DC to await a Regional train to NY. And upon arrival in NY, assuming that the CA is still open, you could gain entry to NY's CA to await a LIRR train or an NJT train if you so desired to wait in the Club Acela. Note: you won't get announcements for either the LIRR or NJT, although NJT trains still appear on the monitors in the CA. LIRR trains don't show on the CA monitors.