This might not fit your actual use case, but instead of creating a single JSON string in jq, just use a shell command group.

echo '{"foo":"bar", "baz":[1,2,3]}' | { echo "My value is:"; jq .; echo "Some other stuff"; }

As @jq170727 mentioned, postprocessing after a pretty-printing run of jq (e.g. jq .) is worth considering. In that vein, here is an awk script that might suffice:

#!/bin/bash

awk '
  function ltrim(x) { sub(/^[ \t]*/, "", x); return x; }
  s && NF > 1 && $NF == "["  { s=s $0;               next}
  s && NF == 1 && $1 == "]," { print s "],";   s=""; next}
  s && NF == 1 && $1 == "["  { print s;        s=$0; next}
  s && NF == 1 && $1 == "{"  { print s; print; s=""; next}
  s && NF == 1 && $1 == "]"  { print s $1;     s=""; next}
  s && NF == 1 && $1 == "}"  { print s;        s=$0; next}
  s                          { s=s ltrim($0);        next}
  $NF == "["                 { s=$0;                 next}
  {print}
'

Examples

With the example input, the invocation:

 jq . example.json | ./pp

produces:

{
  "frameGrid": {
    "size": [24,24],
    "dimensions": [1,1],
    "names": [
      ["default"]
    ]
  }
}

The invocation:

jq -n '{a:[1,2,3,[1,2,3,4]],b:2,c:{d:[1,2,{e:[3,4]}]}}' | ./pp

produces:

{
  "a": [1,2,3,
    [1,2,3,4]
  ],
  "b": 2,
  "c": {
    "d": [1,2,
      {
        "e": [3,4]
      }
    ]
  }
}