@takeshin's answer is fine if you have access to the build.xml file, but this may break, especially if you are building on a slave node (since the slave does not have the referenced build.xml).
Fear not, since you can access this information via Jenkins directly, using its remote access api:
https://wiki.jenkins-ci.org/display/JENKINS/Remote+access+API
For example:
http://<host>/jenkins/job/<job_name>/lastSuccessfulBuild/api/xml
(will give you the xml content... you could replace xml with json to get json content back instead of XML, for example).
NOTE that you may need to use authentication if you've setup your Jenkins instance to require it. Again, fear not: https://wiki.jenkins-ci.org/display/JENKINS/Authenticating+scripted+clients
Then it's a simple matter of parsing the XML for what you want. Something like this, perhaps:
curl --silent --user $USER:$API_TOKEN $URL | grep "<lastBuiltRevision>" | sed 's|.*<lastBuiltRevision>.*<SHA1>\(.*\)</SHA1>.*<branch>.*|\1|'
So, pulling it all together, you can end up with a (relatively) simple shell script to retrieve the last good revision hash from Jenkins:
#!/bin/sh
GIT_LOG_FORMAT="%ai %an: %s"
USER=<username>
API_TOKEN=<api_token>
LAST_SUCCESS_URL_SUFFIX="lastSuccessfulBuild/api/xml"
#JOB_URL gets populated by Jenkins as part of the build environment
URL="$JOB_URL$LAST_SUCCESS_URL_SUFFIX"
LAST_SUCCESS_REV=$(curl --silent --user $USER:$API_TOKEN $URL | grep "<lastBuiltRevision>" | sed 's|.*<lastBuiltRevision>.*<SHA1>\(.*\)</SHA1>.*<branch>.*|\1|')
# Pulls all commit comments since the last successfully built revision
LOG=$(git log --pretty="$GIT_LOG_FORMAT" $LAST_SUCCESS_REV..HEAD)
echo $LOG
Cheers,
Levi