Context. Alma Cycle 3 observations serendipitously showed strong absorption from diffuse molecular gas in the Galactic bulge at − 200 km s⁻¹ < v < −140 km s⁻¹ toward the compact extragalactic continuum source J1744-3116 at (l, b) = −2.13°, − 1.00°. Aims. We aimed to test whether molecular gas in the bulge could also be detected toward the three other, sufficiently strong mm-wave continuum sources seen toward the bulge at |b| < 3°. Methods. We took absorption profiles of HCO⁺ (1−0), HCN(1−0), C₂H (1−0), CS(2−1) and H¹³CO⁺ (1−0) in ALMA Cycle 4 toward J1713-3418, J1717-3341, J1733-3722 and J1744-3116. Results. Strong molecular absorption from disk gas at |v|≲30 km s⁻¹ was detected in all directions, and absorption from the 3 kpc arm was newly detected toward J1717 and J1744. However, only the sightline toward J1744 is dominated by molecular gas overall and no other sightlines showed molecular absorption from gas deep inside the bulge. No molecular absorption was detected toward J1717 where H I emission from the bulge was previously known. As observed in HCO⁺, HCN, C₂H and CS, the bulge gas toward J1744 at v < −135 km s⁻¹ has chemistry and kinematics like that seen near the Sun and in the Milky Way disk generally. We measured isotopologic ratios N(HCO⁺)/N(H¹³CO⁺) > 51 (3σ) for the bulge gas toward J1744 and 58 ± 9 and 64 ± 4 for the disk gas toward J1717 and J1744, respectively, all well above the value of 20−25 typical of the central molecular zone. Conclusions. The kinematics and chemistry of the bulge gas observed toward J1744 more nearly resemble that of gas in the Milky Way disk than in the central molecular zone.