You can try like below. You were almost good, missing grid-auto-flow:dense; to allow the item to fill all the spaces.

.grid-container {
  padding: 20px;
  display: grid;
  grid-gap: 20px;
  grid-auto-rows: 1fr;
  grid-template-columns: repeat(2, 1fr);
  grid-auto-flow:dense;
  counter-reset: albumList;
}

.item {
  aspect-ratio: 1;
  background: #ccc;
  display: flex;
}

/* Number */
.item:before {
  counter-increment: albumList;
  content: counter(albumList);
  margin: auto;
  font-size: 40px;
  color: #000000;
}

@media screen and (min-width: 40em) and (max-width: 63.99875em) {
  /* 640 ~ 1023 */
  .grid-container {
    grid-template-columns: repeat(3, 1fr);
  }
  .item:nth-child(6n + 1),
  .item:nth-child(6n + 6){
    grid-area: span 2/span 2;
  }
  .item:nth-child(6n + 5) {
    grid-column: 1;
  }

}

@media print, screen and (min-width: 64em) {
  /* 1024+ */
  .grid-container {
    grid-template-columns: repeat(4, 1fr);
  }
  .item:nth-child(10n + 1),
  .item:nth-child(10n + 10){
    grid-area: span 2/span 2;
  }
  .item:nth-child(10n + 8) {
    grid-column: 1;
  }
  .item:nth-child(10n + 9) {
    grid-column: 2;
  }
}

<div class="grid-container">
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
  <div class="item"></div>
</div>