With the round fork blades, track ends, drop bar, I suspect this may have been an entry level track bike originally. It has many details reminiscent of one I used to have, a Velox "Alvin" from the 1930's.
I think some are getting a little carried away here.
Rear facing dropouts do not indicate a track bike.
The coaster brake, tyres, mudguards.... and department store origin point to being a roadster. It was a bike, for transport. It was ridden to work, shops etc.
08 Aug 1940 - Advertising - Trove
I think Ms Oliver would have been on a ladies step-through bike. If the bike was hanging in the garage for 30 years, then I'd be surprised as to why June Oliver's bike riding featured in the short obituary. I would have thought the obit would seek to resonate with current members and provide a balance with old and recent.
I would think the tyres were 28- 1 3/8".
Are there any engravings evident on the chainring or crank arms?
To my eye the headtube decal has a curve leading to the section that is missing. The rainbow and wheel graphic would suggest it is original.
Have you asked family members if they have any information on the bike?