A scheme of modular competitive immunochromatography with an analyte-independent test strip and changeable specific immunoreactants has been proposed. Native (detected) and biotinylated antigens interact with specific antibodies during their preincubation in solution, that is, without the immobilization of reagents. After this, the detectable complexes on the test strip are formed by the use of streptavidin (which binds biotin with high affinity), anti-species antibodies, and immunoglobulin-binding streptococcal protein G. The technique was successfully applied for the detection of neomycin in honey. The visual and instrumental detection limits were 0.3 and 0.014 mg/kg, respectively, and the degree of neomycin revealed in honey samples varied from 85% to 113%. The efficiency of the modular technique with the use of the same test strip for different analytes was confirmed for streptomycin detection. The proposed approach excludes the necessity of finding the condition of immobilization for each new specific immunoreactant and transferring the assay to other analytes by a simple choice of concentrations for preincubated specific antibodies and the hapten–biotin conjugate.