I recently had a requirement to change our Jenkins URL from http://our.webdomain:8080 to http://our.webdomain:8080/jenkins. The application is installed in Ubuntu. I tried to read the script at /etc/init.d/jenkins and was able to find out that this can be easily done by modifying /etc/default/jenkins.
The solution is just to add --prefix=$PREFIX or --prefix=/jenkins to JENKINS_ARGS.
# servlet context, important if you want to use apache proxying
PREFIX=/jenkins
# arguments to pass to jenkins.
# --javahome=$JAVA_HOME
# --httpPort=$HTTP_PORT (default 8080; disable with -1)
# --httpsPort=$HTTP_PORT
# --ajp13Port=$AJP_PORT
# --argumentsRealm.passwd.$ADMIN_USER=[password]
# --argumentsRealm.roles.$ADMIN_USER=admin
# --webroot=~/.jenkins/war
# --prefix=$PREFIX
JENKINS_ARGS="--webroot=/var/cache/jenkins/war --httpPort=$HTTP_PORT --ajp13Port=$AJP_PORT --prefix=$PREFIX"
