We report measurements of the inclusive transverse momentum (pT) distribution of centrally produced K0S, K⋆±(892), and ϕ0(1020) mesons up to pT=10 GeV/c in minimum bias events, and K0S and Λ0 particles up to pT=20 GeV/c in jets with transverse energy between 25 and 160 GeV in pp¯ collisions. The data were taken with the CDF II detector at the Fermilab Tevatron at s√=1.96 TeV. We find that as pT increases, the pT slopes of the three mesons (K0S, K⋆±, and ϕ) are similar. And using our previous Λ0 results from minimum bias events, we show that the ratio of Λ0 to K0S as a function of pT in minimum bias events becomes similar to the fairly constant ratio in jets at pT∼5 GeV/c. This suggests that the particles with pT≳5 GeV/c in minimum bias events are from “soft” jets, and that the pT slope of particles in jets is insensitive to light quark flavor (u, d, or s) and to the number of valence quarks. We also find that for pT≲4 GeV relatively more Λ0 baryons are produced in minimum bias events than in jets.