Para fazer o Alpine usar a mesma pasta Enviados usada por outros clientes de e-mail, você deve definir a pasta Fcc (File carbon copy) (na qual as cópias Alpine enviam mensagens) para uma pasta no seu Servidor IMAP. Algo parecido com isto:
{imap.host.com:993/ssl/novalidate-cert/[email protected]}Sent
Veja a documentação da Alpine:
This value specifies where a copy of outgoing mail should be saved. If this is not a path name, it will be in the default collection for saves. Any valid folder specification, local or IMAP, is allowed. This default folder carbon copy only applies when the "Fcc Name Rule" is set to use the default folder. Unix Alpine default is normally "sent-mail" in the default folder collection.
If you access your email through an IMAP server, especially if you often switch between Unix Alpine and PC-Alpine, or between various desktop email systems, you may want to set this to a folder on your IMAP server (remember that in order to later access this remote folder through Alpine, it must be in a folder collection. See Folder Collections Explained for more information).