A flexible homopolymer chain with sufficiently short-range interactions undergoes a discontinuous transition from an expanded coil to a compact crystallite analogous to the all-or-none folding transition exhibited by fast-folding proteins. In this work we investigate the effects of both tethering and confinement on this type of folding transition. In particular, we study a flexible square-well N-mer chain (monomer diameter sigma) located between two hard walls forming a slitlike pore (width W) with the chain end tethered to one wall. We carry out Monte Carlo simulations with Wang-Landau sampling to construct the single-chain density of states and use both microcanonical and canonical analyses- to characterize phase transitions. We find that in a very wide pore (W > N sigma) the all-or-none folding transition is only slightly affected by the tethering. As the pore width is reduced to near the size of the folded polymer chain (W approximate to N-1/3 sigma) there is a modest entropy reduction for the unfolded states leading to a stabilization of the folded state. Below a critical slit width the chain is unable to fold into the native ground state. However, discontinuous all-or-none folding still occurs to higher energy ground-state structures commensurate with the slit width. All-or-none folding persists even to the limit of a very narrow pore (W approximate to sigma) where the ground state structure is a quasi-two-dimensional crystal. Both structural and thermodynamic effects of confinement are studied including quantitative analyses of entropy reduction and changes to the free energy barrier to folding.