Bungee or travel straps

Look on Amazon for luggage straps. There are different styles. Some like this.

https://www.amazon.com/Luggage-Suitcase-Adjustable-Applications-Black-001/dp/B07QB1J3N6/ref=sr_1_1?keywords=luggage%2Bstraps&qid=1690742377&refinements=p_89%3AZINZ&rnid=2528832011&s=apparel&sr=1-1&th=1

Try one with a trolley sleve that secures the day bag to the handle.

Yes. There are also just add-a-bag strapa that loop the handle of the smaller bag to the handle of the larger bag and carries it on the front instead of on the top.
I have a rolling garment bag that has an add-a-bag strap built in.

I've used just regular bungee cords, although it might depend on the size and/or shape of your day bag.

I sewed a trolley sleeve onto the back of my under-seat bag so that it would stay on top of my roller bag. I also had to give up on my beloved orange carry-on roller bag due to its single post handle --- I tried every way I could think of to make my under-seat bag stay on. Now I have a boring gray bag with double posts.

Use bunghee(s). Why? Because they have other travel uses. For example, laundry clothes line, belt, secure a door, and bundle firewood.

I have the travelon bungee. It does stretch to some degree, but it streI've used it on trips, where needed. It works fine. It securely holds a personal item or even just coats. I do have to wait until I have cleared the jetway before attaching it, as it does take a few seconds; That would be my only complaint.

When I have a personal item and a carryon, I use the add a bag straps that come with two of my regular bags. This is my preferred way so I look for personal items that are less than 16" tall that have a small tote handle. This system makes my bags almost waitless. It's the system flight attendants use, so it must be practical. As I recall, it fits all my bags so far and is adaptable.

PS: if you want me to test an approximate size personal item on a wheeled carryon, I'm happy to do that if I have the right size...which considering my connection I am likely to have it. I just need to dig it out of the closet. Let me know if you do and give me measurements of the item you want to attach.

Hi Wray, Thanks so much for your offer to test the bungee. I have a Travelpro bag that is 18" wide x 11" high x 8" deep. It has the rear "strap" with velcro that allows for slipping it onto the handle of the larger roller bag. Again, thanks so much for your offer to "test" for me!

Product tester here. LOL. To mimic your personal item, the best I could do was an 18" X 14" X 8" (Ebags Motherlode Jr.). I threw stuff in it so that the weight was on the bottom of the bag (the bottom of the 18") length. If you pack evenly, your results might be less so. The bungee cord wrapped easily around this item and then I toggled it tight as directed by the directions. I live in a multi-unit building so I have a long common corridor, carpeted, so, much like many airports. When walking straight and normally the bag stayed on just fine. If I started swinging my arm behind me, the arm holding the suitcase handle, repeatedly back and forth perhaps a foot or so, all the way down the corridor, the bag started slipping so that the weighted end of my personal item started to slip toward the weighted side and off balance...after the long walk of doing this. I was not good enough to go outside and drag it off curbs (sorry). I imagine a few curbs would be fine. Dragged down a set of stairs, I'm not sure it wouldn't become unbalanced packed the way I packed it. The bag that held the personal item was an 21 " Eagle Creek Tarmac which has very little solid top...5 1/2 inches deep by 10 inches wide, after that the bag curves away from this base. I thought this might be the issue. So I pulled out my husband's Studio H Hartmann bag, which has a solid, flat top measuring 14" wide by 7 1/2" deep. It only took two shakes to knock the personal item (still my Ebags BP) fully off balance and falling. Frankly, I'm surprised on this result.

Conclusion: It depends what kind of bag you are using for the base. I know years ago I used this bungee on a Rick Steves 21" wheeled bag and had no issues, but my personal item was not as large as your bag. It also works great for 2 winter coats while in the airport as I recall. Feel free to ask other questions. I hope this helps.

PS, If you personal bag has the fabric sleeve to put it over the handle, then you won't have an issue with it falling off balance as I had.

After struggling with balancing a personal item or putting a strap around it and my carryon handle for years, I changed my personal item to one that had a trolly/luggage sleeve this year. I did the same thing for 2 granddaughters too, we couldn’t be happier. I got a Baggallini backpack with sleeve. I got Osprey backpacks with sleeves for the granddaughters. I even got new foldable, packable duffle bags for all of us. We’ve all taken long international trips this summer and are really happy with the results!

Daiso has a strap that works well. There’s a velcro end that wraps around the suitcase handle and then it stretches up over the additional bag and loops onto the handle of the suitcase. I’ve used it on two multi week trips and it’s been great and held securely.

Edited to add: unfortunately, I’m not finding it online. I’ve found the best time to get travel related items at Daiso is around springtime, so this item may reappear by then, but perhaps too late for your next trip.