The exclusive reaction γp→pK+K− was studied in the photon energy range 3.0–3.8 GeV and momentum transfer range 0.6<−t<1.3 GeV2. Data were collected with the CLAS detector at the Thomas Jefferson National Accelerator Facility. In this kinematic range the integrated luminosity was approximately 20 pb−1. The reaction was isolated by detecting the K+ and the proton in CLAS, and reconstructing the K− via the missing-mass technique. Moments of the dikaon decay angular distributions were extracted from the experimental data. Besides the dominant contribution of the ϕ meson in the P wave, evidence for S−P interference was found. The differential production cross sections dσ/dt for individual waves in the mass range of the ϕ resonance were extracted and compared to predictions of a Regge-inspired model. This is the first time the t-dependent cross section of the S-wave contribution to the elastic K+K− photoproduction has been measured