In this study, we present an attempt to use low earth orbit (LEO) satellites as an alternative when the global navigation satellite system (GNSS) is unavailable for positioning because of spoofing or jamming attacks. Specifically, we design a GNU radio-based SDR receiver that supports the multi-constellation of Orbocom and Iridium Next satellites and receives precise Doppler frequencies from the satellites when externally connected to a precision clock oscillator such as a rubidium oscillator. The frequency offsets can be removed from the receiver side using a precision clock oscillator. Further, we incorporated the concepts of transit originated and Doppler positioning and hinted at fingerprint position estimation and map-matching. We propose a unique and novel positioning method by incorporating all these techniques. Using the method, we successfully determined the approximate location within several hundred meters. Stationary receiver positioning results showed an accuracy of approximately 463 m, an improvement of approximately 62% over single constellation positioning.