Problem & Solution!
Maybe this information is also of interest for other Smartermail administrators:

My config:
- Windows 2008 R2 Server / IIS 7
- SmarterMail 16
- Lets Encrypt Certificate installed with CertifyTheWeb
 
After installing the Lets Encrypt certificates, I noticed that several test pages (digicert, checktls, etc.) detected an error. The tools reported that the SSL chain is not working properly. More detailed research showed that there is a problem between the Root (X3) and the Intermediate Certificate (X1). This seems to be a known problem with IIS servers and Lets Encrypt certificates.
Interestingly, there were no problems with port 443 (webmail), but always when it came to ports served by Smartermail (25,143,587,993 etc).

After a loooong search and reading a lot of threads, I found the ultimate (and very easy) solution! Read it yourself in this forum (Post by Knagis in April 2016):
https://community.letsencrypt.org/t/iis-8-5-building-incorrect-chain-with-lets-encrypt-authority-x3/13320/84
 
After this little intervention I had no more SSL chain problems! :-)
 
-- Dave