A month ago I decided to write a tool which can automate such processes. Starting off with something where a user can configure "our JIRA instance" and "our Jenkins instance", halfway through I switched to a generic approach which can simply connect REST APIs.
I have had Isthmus running internally the last 10 days, and when I commit a change to GIT, the associated JIRA task gets a 'code-change' label. Once Jenkins has deployed the change, the JIRA task gets a 'deployed' label. I find the approach pretty exciting because Isthmus can connect any two REST APIs.
Today I've linked the webpage https://isthmus.want.ch, and put a first version of Isthmus online. Currently I'm asking for a one-time fee, charged is the license key is still in use 30 days after download. I'm not entirely confident about that approach, but maybe I'll get some feedback here.