I already have email setup for one domain, do I setup DNS to send mail using the same server for another?

I already have my DNS records setup for domain1.com. When I use my mail server to handle mail for domain2.com, the mail I send gets rejected by a lot of recipients.

My email server is mail.domain1.com.

mail A 192.0.2.16
mail AAAA 2001:db8::2:16

My SPF record for domain1.com looks like:

@ TXT "v=spf1 IP4:192.0.2.16 IP6:2001:db8::2:16 -all"

My MX record is configured as follows:

@ MX 10 mail.domain1.com.

When I telnet mail.domain1.com 25 and type EHLO, My email server's SMTP banner says:

220 mail.domain1.com ESMTP Postfix (Ubuntu)

If I dig -x 192.0.2.16 I get:

16.2.0.192.in-addr.arpa. 86400 IN PTR mail.domain1.com.

If I dig -x 2001:db8::2:16 I get:

6.1.0.0.2.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.b.d.1.0.0.0.2.ip6.arpa. 86400 IN PTR mail.domain1.com.