If you end up re-siding it's an opporunity moment for tightening up the house and bringing the energy performance up to current code. If it's a 2x4/R11-R13 type of wall, blowing cellulose into the cavities (compressing the batts) the drilling from the exterior gets you part way there. Blown fiber will stuff/plug a lot of the air leakage points, and the higher density increases the air-retardency to reduce air flow rates even if the wall leaks a bit of air. Using a fully adhered weather resistant barrier (eg Henry Blueskin, Vycor enV-S, etc) rather than a stapled-on housewrap will tighten it up even further. The adhered housewrap is more important if the wall is sheathed with ship-lap or other planking. If it's sheathed with plywood that's still in good shape, sealing the seams with purpose-made tapes is almost as good.
To further bring the R-value up to current IRC or NY code (=R20/2x6, or R13 2x4 + R5 continuous insulation) with the thinnest possible increase in profile would be to add a layer of 3/4 " rigid foil-faced polyisocyanurate foam board, taping the seams with the appropriate aluminium tapes. An aluminum facer adjacent to the air gap behind vinyl or aluminum siding adds another R1 of net performance (even though that R1 doesn't count toward code prescriptives). A layer of 3/4" foam board is usually d0-able without major reworking of the window installation, but pay attention to how the window flashing is lapped with the house wrap, or if it can be extended to the exterior side of the foam board. If the flashing can't be lapped to the exterior of the housewrap it may be better to use tapes for air sealing the structural sheathing (only if plywood or OSB, not plank), and use a crinkle-type drainable housewrap (eg Tyvek Drain wrap), or even a mesh type drainable underlayment so that any bulk water directed to the housewrap can drain. Foam-board tight to a flat housewrap on plywood isn't going to cut it if the window flashing directs water between the housewrap & foam board (especially in coastal areas subject to wind-driven rain). But it's fine if the flashing directs bulk water between the foam board and vinyl or aluminum siding, where it drains & dries very easily.