The overtone and multi-mode RR Lyrae stars in the globular cluster M3 are studied using a 200 day long, B,V, and l_c_ time-series photometry obtained in 2012. 70% of the 52 overtone variables observed show some kind of multi-periodicity (with additional frequency at f_0.61_=f_1O_/0.61 frequency ratio, Blazhko effect, double/multi-mode pulsation, and period doubling). A signal at the 0.587 frequency ratio to the fundamental-mode frequency is detected in the double-mode star, V13, which may be identified as the second radial overtone mode. If this mode identification is correct, then V13 is the first RR Lyrae star showing triple-mode pulsation of the first three radial modes. Either the Blazhko effect or the f_0,61_ frequency (or both of these phenomena) appears in seven double-mode stars. The P_1O_/P_F_ period ratio of RRd stars showing the Blazhko effect are anomalous. A displacement of the main frequency component at the fundamental mode with the value of modulation frequency (or its half), is detected in three Blazhko RRd stars that are parallel with the appearance of the overtone-mode pulsation. The f_0.61_ frequency appears in RRc stars that lie at the blue side of the double-mode region and in RRd stars, raising the suspicion that its occurrence may be connected to double-mode pulsation. The changes of the Blazhko and double-mode properties of the stars are also reviewed using the recent and archive photometric data.