We present the first very long baseline interferometric (VLBI) observations of the blazar OJ 287 carried out jointly with the Global Millimeter VLBI Array (GMVA) and the phased Atacama Large Millimeter/submillimeter Array (ALMA) at 3.5 mm on 2017 April 2. The participation of phased ALMA has not only improved the GMVA north-south resolution by a factor of ∼3, but has also enabled fringe detections with signal-to-noise ratios up to 300 at baselines longer than 2 Gλ. The high sensitivity has motivated us to image the data with newly developed regularized maximum likelihood imaging methods, revealing the innermost jet structure with unprecedentedly high angular resolution. Our images reveal a compact and twisted jet extending along the northwest direction, with two bends within the inner 200 μas, resembling a precessing jet in projection. The component at the southeastern end shows a compact morphology and high brightness temperature, and is identified as the VLBI core. An extended jet feature that lies at ∼200 μas northwest of the core shows a conical shape, in both total and linearly polarized intensity, and a bimodal distribution of the linear polarization electric vector position angle. We discuss the nature of this feature by comparing our observations with models and simulations of oblique and recollimation shocks with various magnetic field configurations. Our high-fidelity images also enabled us to search for possible jet features from the secondary supermassive black hole (SMBH) and test the SMBH binary hypothesis proposed for this source.