Thanks for the additional details. I think you are on the right track. Once you have separated the Orig features into two sets, one is covered by the Name features, another not, you should be able to use Select Layer By Location to select the Closest (or Share a line segment) Orig_inName by Orig_outName. Then Dissolve them with the appropriate Stat. Fields settings to get the attributes from Orig_inName onto Orig_outName features. That should work.
In a slightly different way, I have tested the following workflow, assuming the following feature class names:
fc_grdCode (the one with GRIDCODE)
fc_needAttr (the one that will get attributes from fc_GrdCode)
1. Use Union to union fc_grdCode with fc_needAttr. Let's call it fc_grdCodeUnion.
2. Use Select Layer By Attribute to select fc_grdCodeUnion, where FID_fc_grdCode = -1. These are the polygons not covered by fc_grdCode.
3. Use Delete Features tool to delete the selected features. What's left are the polygons covering fc_needAttr, but some of them have not received the grid code.
4. Use Select Layer By Attribute to select fc_grdCodeUnion by GRIDCODE = 0; these are the polygons outside fc_grdCode.
5. Use Select Layer By Location to select fc_grdCodeUnion by the selected features from step 4 that "share a line segment with the source feature". Now you should have the some polygons in fc_grdCodeUnion selected, the ones that you want to merge and get the desired attributes.
6. Run Dissolve with the FID_fc_needAttr field as the Dissolve Field; specify the desired Stat. Fields, for example for your GRIDCODE, use MEAN or MAX. Do not use FIRST becasue it might grab the one without the grid code value first. Let's say your output is fc_needAttr_dissolved.
7. Since you still have the selection from step 5, run Delete Features to delete them from grdCodeUnion.
8. You can now use Append tool to combine the remaining features in grdCodeUnion with fc_needAttr_dissolved, specifying the desired Field Map. I think the output is what you want.
Depending on your data, if you have disjoint polygons as shown in your image, you can try CLOSEST selection or Spatial Join to get attribes from the above result. I think you know enough to figure that out.
Hope I didn't make mistakes in my typing. Does that make sense or work for you?