Backup Restore from other node after switchover/failover

Hi,

haven't tested it yet but maybe I can get a quick answer here.
We have discussed the following scenario.

Few nodes as streaming replication cluster all in sync with taking backup
only from one dedicated node. Now that node which is responsible for the backups
goes down. For sure I have a full backup taken from that crashed node
some time ago but now this node is gone and I have to take care for a
restore for some reason.

Am I able to make a full restore with that last backup from the offline node
to any other still existing node with Recovering the archived WAL from the new
selected node if I have activated archive_mode as "always"?

Or is it strictly necessary to create a new full backup from the new node
which is responsible for the backups after the failover or maybe after a
Switchover, too?

Thanks and best regards

Dirk