The number of existing global positioning system (GPS) single-frequency receivers continues growing. More than 90% of GPS receivers are implemented as low-cost single-frequency chipsets embedded in smartphones. This provides new opportunities, in particular for ionospheric sounding. In this context, we present the new sidereal days ionospheric graphic (SIg) combination of single-frequency GNSS measurements. SIg is able to monitor, for each given GNSS transmitter-receiver pair, the vertical total electron content (VTEC) relative to the previous observation with the same or almost the same line-of-sight (LOS) vector. In such arrangements the SIg multipath error mostly cancels, thus increasing the accuracy of the ΔVTEC significantly. This happens for the GPS constellation after one sidereal day (about 23 h 56 m) and for Galileo after 10 sidereal days approximately. Moreover, we show that the required calibration of the corresponding carrier phase ambiguity can be accurately performed by means of VTEC global ionospheric maps (GIMs). The results appear almost as accurate as those based on the dual-frequency technique, i.e., about 1 TECU or better, and with much more precision and resolution than the GIM values in the ionospheric region sounded by each given single-frequency receiver. The performance is demonstrated using actual data from 9 permanent GPS receivers during a total solar eclipse on August 21, 2017 over North America, where the corresponding ionospheric footprint is clearly detected in agreement with the total solar eclipse predictions. The advantages of extending SIg to lower carrier frequencies and the feasibility of applying it to other global navigation satellite system (GNSS) systems are also studied. This is shown in terms of a fully consistent VTEC depletion signature of the same eclipse phenomena, obtained with Galileo-only data in North America at mid and low latitude. Finally the SIg feasibility, including the cycle slip detection, is shown as well with actual mass-market single frequency GPS receivers at mid and high latitude.