Use 60Amp disconnect as a pass-through to another disconnect
This should be fine, but triple-check the fill if you want to be sure, and make sure it's labeled properly too!

Your usecase (extra wires run through a disconnect box to another disconnect) falls under NEC 312.8:


  312.8 Switch and Overcurrent Device Enclosures with
  Splices, Taps, and Feed-Through Conductors. The wiring space of enclosures for switches or overcurrent devices
  shall be permitted for conductors feeding through, spliced, or tapping off to other enclosures, switches, or overcurrent
  devices where all of the following conditions are met:
  
  (1) The total of all conductors installed at any cross section
  of the wiring space does not exceed 40 percent of the
  cross-sectional area of that space.
  
  (2) The total area of all conductors, splices, and taps installed at any cross section of the wiring space does not
  exceed 75 percent of the cross-sectional area of that
  space.
  
  (3) A warning label complying with 110.21(B) is applied
  to the enclosure that identifies the closest disconnecting
  means for any feed-through conductors.


The 40% and 75% fill rules are trivial in this case -- the gutters on even the smallest disconnects are larger than 3/4" EMT in order to provide the Code-required wire-bending space, so you'll be nowhere close to either one.  Since these are true feed-through conductors, though, the existing disconnect will need a a label applied to it in accordance with 110.21(B), though:


  (B) Field-Applied Hazard Markings. Where caution,
  warning, or danger signs or labels are required by this
  Code, the labels shall meet the following requirements:
  
  1) The marking shall adequately warn of the hazard using
  effective words and/or colors and/or symbols.
  
  Informational Note:
  ANSI Z535.4-2011, Product Safety
  Signs and Labels, provides guidelines for suitable font
  sizes, words, colors, symbols, and location requirements for
  labels.
  
  (2) The label shall be permanently affixed to the equipment
  or wiring method and shall not be hand written.
  
  Exception to (2): Portions of labels or markings that are
  variable, or that could be subject to changes, shall be permitted to be hand written and shall be legible. 
  
  (3) The label shall be of sufficient durability to withstand
  the environment involved.
  
  Informational Note: ANSI Z535.4-2011, Product Safety
  Signs and Labels, provides guidelines for the design and
  durability of safety signs and labels for application to electrical equipment.


Given all that, I would simply make up a printed label that says "Caution: Feed-through conductors are disconnected by breaker XYZ in the panel" where XYZ is the label given to the new breaker in the loadcenter's directory, and apply it to the inside of the existing disconnect's cover where it's sheltered from UV, water, etal.