We have studied feasibility of single-layer second-order $high-T_c$ SQUID gradiometers in magnetocardiography. We have measured human cardiomagnetic signals using a short-baseline (5.8 mm) single-layer second-order YBCO gradiometer in partially shielded environments. The gradiometer has an overall size of $17.6;mm{ imes}6;mm$ and contains three parallel-connected pickup coils which are directly coupled to a step-edge junction SQUID. The gradiometer showed an unshielded gradient noise of $0.84;pT/cm^2/Hz^{1/2}$ at 1 Hz, which corresponds to an equivalent field noise of $280;fT/Hz^{1/2}$. The balancing factor was $10^3$. Based on the same design rules as the short-baseline devices, we have studied fabrication of 30 mm-long baseline gradiometers. The devices had an overall size of $70.2;mm{ imes}10.6;mm$ with each pickup coil of $10;mm{ imes}10;mm$ in outer size. As Josephson elements we made two types of submicron bridges, which are variable thickness bridge (VTB) and constant thickness bridge (CTB), from $3;{mu}m-wide$ and 300 nm-thick YBCO lines with a thin layer of Au on top by using a focused ion beam (FIB) patterning method. VTB was 300 nm wide, 200 nm thick, 30 nm long with Au removed and CTB 100 nm wide and 30 nm long. In temperature-dependent critical currents, $I_c(T)$, VTB showed an nonmetallic barrier-type behavior and CTB an SNS behavior. We believe that those characteristics are ascribed to naturally formed grain boundaries crossing the bridges.