We study the process e+e- → π+π- φ (3686) using 5.1 fb-1 of data collected at 16 center-of-mass energy (√s) points from 4.008 to 4.600 GeV by the BESIII detector operating at the BEPCII collider. The measured Born cross sections for e+e- → π+π- φ(3686) are consistent with previous results, but with much improved precision. A fit to the cross section shows contributions from two structures: the first has M = 4209.5 ± 7.4 ±1.4 MeV=c2 and Λ = 80.1 ± 24.6 ± 2.9 MeV, and the second has M = 4383.8 ± 4.2 ± 0.8 MeV=c2 and F = 84.2 ± 12.5 ±2.1 MeV, where the first errors are statistical and the second systematic. The lower-mass resonance is observed in the process e+e- → π+π- φ(3686) for the first time with a statistical significance of 5.8σ. A charged charmoniumlike structure is observed in the π± φ(3686) invariant mass spectrum for data at √s = 4.416 GeV. A fit with an S-wave Breit-Wigner function yields a mass M = 4032.1 ± 2.4 MeV=c2, where the errors are statistical only. However, there are still unresolved discrepancies between the fit model and data. The width of the intermediate state varies in a wide range for different kinematic regions within the data set. Therefore, no simple interpretation of the data has been found, and a future data sample with larger statistics and more theoretical input will be required to better understand this issue.