What’s a way to proxy an https site to an http endpoint?

I’ve got a piece of software that doesn’t support https, only http.

I’m trying to access a secure website from it. I want to create an endpoint at http://localhost:80 to https://securesite.com (443). My software can browse the secure site in cleartext at the local endpoint.

I’ve tried to do this with stunnel, but remote hosts don’t behave right; it seems like they can’t read what the URL path is supposed to be; I’m guessing it’s a headers issue of some kind.

Is there an easier way to do this?

Edit:
Current stunnel config:

[linkly]
accept = 81
client = yes

connect = linklyhq.com:443
CApath = /etc/ssl/certs/
checkHost = linklyhq.com 
OCSPaia = no 
verifyChain=yes