Relativistic jets are observed in some accreting black-hole systems but not others. In X-ray binaries, jet production is closely associated with geometrically thicker accretion states, suggesting that disk thickness may influence the conditions for jet launching. We investigate this connection using three-dimensional GRMHD simulations of disks with h/r=0.1--0.3, initialized with a toroidal magnetic field with plasma β=5. We find that the magnetorotational instability (MRI) generates large-scale poloidal fields through non-axisymmetric shearing waves, with an additional contribution from axisymmetric induction through stretching by channel-like modes. The resulting mean fields become progressively weaker as the disk becomes thinner. Because these fields regulate MRI turbulence, thinner disks exhibit weaker angular-momentum transport and accrete more slowly than the standard v_(rm acc)∝(h/r)² thin-disk scaling would predict. The magnetic-flux transport velocity remains comparable to the accretion velocity, so weaker field generation and slower transport combine to strongly suppress black-hole flux accumulation at smaller h/r. Correspondingly, the thinnest disk (h/r=0.1) produces an electromagnetic jet efficiency of only η_rmEM∼1%, while the thickest disk reaches the magnetically arrested disk (MAD) state and attains η_rmEM≥100%. Using the measured large-scale magnetic flux distribution and transport velocities, we estimate the time required to accumulate MAD-level magnetic flux. Only the h/r∼0.3 disk reaches MAD in the duration of our simulations, at t≃2×10⁴ r_g/c, whereas for h/r=0.1 we estimate a timescale of ∼10⁸ r_g/c. These results link disk geometry to magnetic-flux accumulation and jet launching, with direct implications for the timescales over which jets can emerge in transient accreting systems.