A new search signature for excited leptons is explored. Excited muons are sought in the channel $pp \to \mu\mu^* \to \mu \mu\textrm{ jet jet}$, assuming both the production and decay occur via a contact interaction. The analysis is based on 20.3 fb$^{-1}$ of $pp$ collision data at a centre-of-mass energy of $\sqrt{s}$ = 8 TeV taken with the ATLAS detector at the Large Hadron Collider. No evidence of excited muons is found, and limits are set at the 95% confidence level on the cross section times branching ratio as a function of the excited-muon mass $m_{\mu^*}$. For $m_{\mu^*}$ between 1.3 TeV and 3.0 TeV, the upper limit on $\sigma B(\mu^* \to \mu q \bar{q}$) is between 0.6 and 1 fb. Limits on $\sigma B$ are converted to lower bounds on the compositeness scale $\Lambda$. In the limiting case $\Lambda = m_{\mu^*}$, excited muons with a mass below 2.8 TeV are excluded. With the same model assumptions, these limits at larger $\mu^*$ masses improve upon previous limits from traditional searches based on the gauge-mediated decay $\mu^* \to \mu \gamma$.