Long-baseline interferometry is an important technique to spatially resolve binary or multiple systems in close orbits. By combining several telescopes together and spectrally dispersing the light, it is possible to detect faint components around bright stars. Aims. We provide a rigorous and detailed method to search for high-contrast companions around stars, determine the detection level, and estimate the dynamic range from interferometric observations. We developed the code CANDID (Companion Analysis and Non-Detection in Interferometric Data), a set of Python tools that allows us to search systematically for point-source, high-contrast companions and estimate the detection limit. The search pro- cedure is made on a N x N grid of fit, whose minimum needed resolution is estimated a posteriori. It includes a tool to estimate the detection level of the companion in the number of sigmas. The code CANDID also incorporates a robust method to set a 3{sigma} detection limit on the flux ratio, which is based on an analytical injection of a fake companion at each point in the grid. We used CANDID to search for the companions around the binary Cepheids V1334 Cyg, AX Cir, RT Aur, AW Per, SU Cas, and T Vul. First, we showed that our previous discoveries of the components orbiting V1334 Cyg and AX Cir were detected at > 13 sigmas. The companion around AW Per is detected at more than 15 sigmas with a flux ratio of f = 1.22 +/- 0.30 %. We made a possible detection of the companion orbiting RT Aur with f = 0.22 +/- 0.11 %. It was detected at 3.8{sigma} using the closure phases only, and so more observations are needed to confirm the detection. We also set the detection limit for possible undetected companions. We found that there is no companion with a spectral type earlier than B7V, A5V, F0V, B9V, A0V, and B9V orbiting V1334 Cyg, AX Cir, RT Aur, AW Per, SU Cas, and T Vul, respectively.
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